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\title{Graded rings and birational geometry} \date{}
\author{Miles Reid}
\begin{document}
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\maketitle
\begin{abstract}
This paper is a written version of my lecture ``Rings and varieties'' at
the Kinosaki algebraic geo\-metry workshop in Oct~2000, and a series of two
lectures at Tokyo University in Dec~2000. It is intended to be informative
and attractive, rather than strictly accurate, and I expect it to stimulate
work in a rapidly developing field (as did its predecessor Reid \cite{R3}).
The paper was prepared in a hurry to meet a deadline, and one or two
sections remain in first draft. I apologise to the reader and the referee
for any inconvenience caused.
The canonical ring of a regular algebraic surface of general type, the
graded ring over a K3 surface with Du Val singularities polarised by an
ample Weil divisor, or the anticanonical ring of a Fano variety is a
Gorenstein ring. In simple cases, a Gorenstein ring is a hyper\-surface,
a codimension~2 complete intersection, or a codimension~3 Pfaffian. We
now have additional techniques based on the idea of projection in
birational geometry that produce results in codimension~4 (and 5, etc.),
even though there is at present no useable structure theory for the
graded ring.
This paper applies graded ring methods, especially {\em unprojection\/},
to the existence of Fano 3-folds and of Sarkisov birational links between
them. The 3-fold technology applies also to some extent to construct
canonical surfaces. A recurring theme is that unprojection often acts as a
working substitute for a structure theory of Gorenstein rings in low
codimension. I discuss what little I understand of the structure of
codimension~4 Gorenstein rings, and present a general and entirely useless
structure theorem. The final section of the paper contains a brief outline
of forthcoming joint work with Gavin Brown on $\CC^*$ covers of Mori flips
of Type~A, intended to illustrate the use of serial unprojection.
\end{abstract}
\setcounter{tocdepth}{1}
\tableofcontents \bigskip
\section{Introduction}\label{sec!intro}
On the geometric side, I am interested in the following problems:
\begin{enumerate}
\item Existence of Fano 3-folds
\item Sarkisov birational links between Fano 3-folds
\item Applications of 3-fold technology to canonical surfaces
\item Structure theory of Gorenstein rings in low codimension
\item $\CC^*$ covers of Mori flips.
\end{enumerate}
Question~1 is biregular as stated, but is frequently studied in birational
terms, notably by {\em projection} methods. As a first introduction to this
idea, I spend some time in Section~\ref{sec!Bx-Ay} below on the trivial
algebraic trick
\[
(Bx-Ay=0) \quad\mapsto\quad (xs=A, ys=B)
\]
that goes from a hypersurface to a codimension~2 complete intersection
(c.i.), contracting a divisor $x=y=0$. This has many applications to
constructing new Fano varieties, and links between them. As described in
Papadakis--Reid \cite{PR} and in \ref{ssec!Bx-Ay}--\ref{ssec!CM}, all the
quadratic involutions of Corti--Pukhlikov--Reid \cite{CPR} and most of the
construction of links in Corti and Mella \cite{CM} are here.
The main methods of constructing Fano 3-folds are:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item Graded ring methods,
\item Birational methods,
\item Embedding a variety in a symmetric space in the style of Mukai.
\end{enumerate}
Method (a) is closely related to the question of projective embeddings. On
the algebraic side, the simplest cases are graded rings in low codimension
with a known structure: hypersurfaces, codimension~2 c.i.s, and
codimension~3 Pfaffians.
Method~(c) is currently a distantly perceived aspiration: we hope that we
can eventually understand the usually complicated system of equations
defining a variety in geometric terms, for example, as a section of a {\em
key variety} having an interpretation, say in terms of linear algebra or
algebraic groups. The key variety is often simpler and has more structure
than its lower dimensional sections. In this vein,
Examples~\ref{exa!Go3}--\ref{exa!Ci} obtain several classic and modern
constructions of surfaces and 3-folds as general sections of bigger ``key
varieties''. This is perhaps a model for applications of 3-fold techniques
to older branches of geometry, such as canonical surfaces.
\begin{defn}\label{defn!Fano}
A {\em Fano $3$-fold} (also $\QQ$-Fano 3-fold) is a variety $X$ in the Mori
category (that is, $X$ is projective and has at worst $\QQ$-factorial terminal
singularities) with $\rho=\rank\Pic X=1$ and $-K_X$ ample. The {\em
anticanonical ring} of $X$ is
\[
R=R(X,-K_X)=\directsum_{n\ge0} H^0(\Oh_X(-nK_X)).
\]
It is known to be a Gorenstein ring (see for example \cite{GW}).
An important case is when $-K_X$ generates the class group of $X$, that
is, $\Cl X=\ZZ\cdot(-K_X)$, corresponding to Fano's {\em variet\`a di prima
specie}. The alternative is that $-K_X$ is divisible in $\Cl X$, or that
$\Cl X$ has a finite torsion subgroup (for example, if $X$ is an
Enriques--Fano variety); it is then normally more efficient to work with
the slightly bigger ring $\directsum_{D\in\Cl X} H^0(\Oh_X(D))$, which is
graded by $\NN\oplus\hbox{torsion}$.
\end{defn}
See \cite{CPR}, 3.1 for the definition of Sarkisov link of type~II and
Corti \cite{Co} for more general Sarkisov links. As explained there, a link
$X\broken X'$ of type~II between Fano 3-folds involves first making an
extremal extraction $Y\to X$ (usually a point blowup), then running a
2-ray game or minimal model program on $Y$ until it finally contracts a
divisor back down. All the links constructed in \cite{CPR} are made by
calculating the anticanonical ring of $Y$. That is, the birational
question is attacked by biregular or graded ring methods.
\subsection{Projection through the ages}\label{ssec!proj}
It is interesting that successive generations of algebraic geometers
inter\-pret {\em projection} in several remarkably different ways.
\begin{enumerate}
\renewcommand{\labelenumi}{(\roman{enumi})}
\item Historically, projection always means linear projection of a variety
in (unweighted) projective space $\PP^n$ to a smaller $\PP^{n'}$.
Projection from a general linear centre disjoint from the variety is used
in proving foundational results such as Noether normalisation, or the
existence of a birational projection to a hypersurface, or to define the
dimension, function field or canonical divisor of a variety.
\item Generic projection allows us to assert that any variety has a
birational morphism to a hypersurface with ordinary singularities. {\em
Italian projection} is a technique for studying a canonical surface $S$
that goes back to Enriques and was later developed by Ciliberto \cite{Ci}
and Catanese \cite{Ca}--\cite{Ca2} and others. In modern terms, it
consists of analysing the canonical ring $R(S,K_S)$ of $S$ as a module
over a polynomial ring $k[x_1,\dots,x_n]$ (preferably with $n=4,5$, etc.),
where $x_i\in H^0(K_S)$ or $H^0(2K_S)$ correspond to some initial set of
generators that define a generic projection $X\to\Xbar\subset\PP^3$ or
$\PP^4$. For surfaces with $p_g=4$, taking a basis of $H^0(K_S)$ as
generators (the {\em $1$-canonical map}) is so natural and instinctive
that it is often not perceived as a choice. If $X$ is a canonical surface
then $K_X=\Oh_X(1)$, and the image $\Xbar$ of a projection
$X\to\Xbar\subset\PP^3$ has $K_{\Xbar}=\Oh(d-4)$, so that the difference
between $K_X$ and $K_{\Xbar}$ has to be accounted for by the
normalisation of $\Xbar$ along its double locus (called ``subadjunction''),
the intersection of $\Xbar$ with the adjoint of smallest degree. See
\cite{Ca}--\cite{Ca2} for details.
While in the hands of the maestri this method gives very interesting
examples and results, it is conceptually messy and computationally
unpleasant, and probably intractable. See Problem~\ref{prob!it} for a
comparison between Italian and Gorenstein projection.
\item Meanwhile, del Pezzo exploited linear projections $S_d\broken
S_{d-1}$ between del Pezzo surfaces from a point $P\in S_d$, and Fano and
later Iskovskikh worked with linear projection of a Fano 3-fold $V$ from a
centre contained in $V$, notably projection from a line $\pi_L\colon
V\broken V'$. These take one anticanonical variety to another, that is,
\[
K_V=\Oh_V(-1) \quad\hbox{and}\quad K_{V'}=\Oh_{V'}(-1)
\]
and are cases of {\em Gorenstein projection}: for this to work involves
the discrepancy of the blowup coinciding exactly with the multiplicity of
the centre subtracted from the linear system (compare \cite{PR}, 2.7).
\item Mori and his followers (notably Takeuchi and Takagi) reworked Fano
and Iskovskikh's study of Fano 3-folds in terms of extremal rays or MMP.
Instead of just doing the linear projection that comes instinctively to
someone versed in projective geometry, Mori views Fano's projection
$V\broken V'$ as first the blowup of a line $\wV\to V$, followed by a
MMP or 2-ray game in the Mori cone of $\wV$, that finds and contracts
extremal rays to obtain first a flop, then a divisorial contraction to $V'$.
\item My view of projection is based on the work on Sarkisov links in
\cite{CPR}: if $X$ is a Fano \hbox{3-fold}, and $X_1\to X$ a Mori
extraction (usually a point blowup), say with exceptional divisor $E$ of
discrepancy $\frac{1}{r}$, the anticanonical ring $R_1=R(X_1,-K_{X_1})$ is
a subring of $R=R(X,-K_X)$, consisting of forms of degree $d$ vanishing
$\frac{d}{r}$ times on $E$ (see Example~\ref{exa!Se1} for a particular
case, and compare \cite{CPR}, 3.4). This ties in closely both with Fano
projections and with the Mori 2-ray game, but in general, it does not
directly predict anything about the algebra of $R_1$ or the geometry of
$Y=\Proj R_1$, or the rational map $X\broken Y$. In good cases, $X\broken
Y$ may be a projection from one weighted projective space (w.p.s.)\ to
another, obtained by eliminating a single generator of $R$ of high weight;
but we do not start out by assuming that, and more complicated things
happen in applications (see Examples~\ref{exa!Se1}--\ref{exa!Se2}).
\end{enumerate}
In higher codimension, the idea of Kustin--Miller unprojection \cite{KM},
\cite{PR} often acts as a workable substitute for a structure theorem. I
discuss this in Sections~\ref{sec!Unproj1}--\ref{sec!T&J} with some pretty
applications. More complicated unprojections not of Kustin--Miller type,
with exceptional divisor that is not projectively Gorenstein, can be used
to similar effect (see Section~\ref{sec!Unproj2}), even when the algebra is
complicated and not really properly understood. The examples of Type~II
unprojections discussed in Section~\ref{sec!Unproj2} arising from Selma
Alt{\i}nok's work \cite{A} are really nontrivial applications of these
methods.
In Section~\ref{sec!Gor4}, I explain an application of geometric ideas to the
structure theory of rings in codimension~4. Although I state a ``structure
theorem'', the answer is still elusive, and my result is not yet explicit
enough to have any predictive power.
The idea of unprojection is just made for serial use. That is, it can be
used many times over in an inductive way to produce Gorenstein rings of
arbitrary codimension, whose properties are nevertheless controlled by
just a few equations as a new unprojection variable is adjoined.
Section~\ref{sec!MoriA} discusses briefly how this applies to the
$\ZZ$-graded rings over Mori flips (forthcoming joint work with Gavin
Brown).
\subsection{Acknowledgments}
Several items in what follows are derived from conversations with Selma
Alt{\i}nok, Gavin Brown, Alessio Corti, Mori Shigefumi, Mukai Shigeru,
Stavros Papadakis and Takagi Hiromichi, and I refer in several places to
results from Papadakis' forthcoming thesis \cite{P}. I thank Takagi for
providing me with excellent lecture notes. My stay in Japan was generously
supported by Kyoto Univ., RIMS, and I am extremely grateful to Professors
Kawamata, Miyaoka, Mori and Saito Kyoji for invaluable assistance and
friendly hospitality. This paper was written during a short summer
solstice visit to John Cannon's Magma group at the University of Sydney; I
thank them for the invitation, and for all the wonderful meals.
\section{The $Bx-Ay$ argument}\label{sec!Bx-Ay}
The most basic example of unprojection consists simply of replacing a
hyper\-surface $Bx-Ay=0$ that contains a codimension 2 c.i.\ $x=y=0$ by
the codimension 2 c.i.\ $xs=A,ys=B$. Despite its trivial appearance, this
trick has many applications.
\subsection{The unprojection variable $s=A/x=B/y$}\label{ssec!Bx-Ay}
Write $\PP=\PP^n(a_0,\dots,a_n)=\Proj k[x_0,\dots,x_n]$ for the w.p.s.\
with weights $\wt x_i=a_i$. Let $D:(x=y=0)\subset\PP^n$ be a codimension~2
c.i.; here $x,y$ could be two of the coordinates $x_i,x_j$, or any two
hypersurfaces with no common components. Then any hyper\-surface
containing $D$ is of the form
\[
X:(Bx-Ay=0)\subset\PP^n(a_0,\dots,a_n).
\]
Assume that $\deg A>\wt x$. Now define
\[
Y:(xs=A,ys=B)\subset\PP^{n+1} = \Proj k[x_0,\dots,x_n,s],
\]
where $\wt s=\deg A-\wt x$. Then $Y$ contains the point ``at infinity'' of
the w.p.s.\ $P_s=(0:\cdots:0:1)$, where $x_i=0$ for all $i$, but $s\ne0$.
There are two inverse birational maps: $X\broken Y$ is the unprojection, or
the graph of $s$, obtained by adjoining the unprojection variable
$$\refstepcounter{subsection}
s =\frac{A}{x} = \frac{B}{y}\,.
\label{eq!s=A/x}
\eqno{(\thesubsection)}
$$
The inverse $Y\broken X$ corresponds algebraically to eliminating $s$. In
terms of geo\-metry, it blows $P_s$ up to a divisor $D\subset X$.
The following familiar setup is a special case of the $Bx-Ay$ trick: let
\[
L\subset S_3\subset\PP^3
\]
be a cubic del Pezzo surface containing the line $L:(x=y=0)$. Then the
defining equation of $S_3$ is $Bx-Ay$, where $A,B$ are quadratic polynomials
in $\PP^3$. The condition for $S_3$ to be nonsingular along $L$ is that $A,B$
have no common zeros on $L$, so that $s$ given by (\ref{eq!s=A/x}) is well
defined, and defines a morphism $S_3\to T_4=Q_1\cap Q_2\subset\PP^4$ to a
del Pezzo surface of degree~4. This is the contraction morphism of $L$
provided by Castelnuovo's criterion.
However, the same equations apply much more generally: the hyper\-surface
$X:(Bx-Ay=0)$ can be of any degree in a w.p.s.\ of any dimension, and can
be arbitrarily singular, provided only that $x,y$ remains a regular sequence.
If $A,B$ do not both vanish along any component of $x=y=0$ (that is, if
$D:(x=y=0)\subset X$ is a Weil divisor, or a Cartier divisor at every generic
point), then $X\broken Y$ is birational.
\subsection{Application to Sarkisov links}\label{ssec!Sark}
Consider an anticanonically embedded hypersurface
\[
\PP(1,a_1,a_2) \subset X_d \subset \PP(1,a_1,a_2,a_3,a_4)
\]
of degree $d=\deg X=a_1+\cdots+a_4$ containing a plane $\PP(1,a_1,a_2)$.
Here $X$ is one of the ``famous 95'', but {\em is not in the Mori
category}: it has equation $Bx_3-Ax_4=0$, and is not $\QQ$-factorial at
points with $A=B=x_3=x_4$.
Assume that $a_4>a_3$. Then $X$ is the {\em midpoint} of a Sarkisov link
of type~II
$$\refstepcounter{subsection}
Z \lbroken X \broken Y,
\label{eq!Slink}
\eqno{(\thesubsection)}
$$
which is either one of the quadratic involutions of \cite{CPR}, 4.4--4.9, or
of the type studied by Corti and Mella \cite{CM}. Both broken arrows are
given by the $Bx-Ay$ trick: suppose that $X$ is the hypersurface
$X:(Bx_3-Ax_4=0)$. The rational map $X\broken Y$ contracts the plane
$\PP(1,a_1,a_2)$ to the point $P_s\in Y$, with $Y$ the graph of
$s=\frac{A}{x_3}=\frac{B}{x_4}$, and
\[
Y_{d-a_3,d-a_4}:(sx_3=A,sx_4=B) \subset \PP(1,a_1,\dots,a_4,a_1+a_2).
\]
This is a {\em general} codimension~2 c.i.\ of the stated degrees. If
$A=x_4$, then the first equation $sx_3=A$ eliminates $x_4$, and $Y$ is a
general hypersurface in $\PP(1,a_1,\dots,a_3,a_1+a_2)$. In this case
$X:(x_4^2+\cdots=0)$ has a biregular involution, $Y\iso Z$, and the link
(\ref{eq!Slink}) is one of the quadratic involutions of \cite{CPR},
4.4--4.9.
On the other hand, $Z$ is the graph of $t=\frac{x_4}{x_3}=\frac{B}{A}$
(recall that $a_4>a_3$). Then $X\broken Z$ contracts the divisor
$D:(x_3=A=0)$, and $Z$ is defined by the equations $x_4=tx_3$, $At=B$.
Because of the first equation, $Z$ is still a hypersurface
\[
Z_{d-a_3}\subset \PP(1,a_1,a_2,a_3,a_4-a_3)
\]
with defining equation $F=A(x_0,\dots,x_3,tx_3)t-B(x_0,\dots,x_3,tx_3)$,
that is, $At-B$ after the substitution $x_4\mapsto tx_3$. Because of this,
$Z_{d-a_3}$ is not a {\em general} hypersurface of the stated degree. It is
a Fano 3-fold in the Mori category, but has a funny terminal singularity
at the point $P_t$. At this point, the classification of Sarkisov links
gets tangled up with the classification of divisorial extractions in the
Mori category, on which there has been considerable recent progress; see
Corti--Mella \cite{CM}, Kawakita \cite{Ka}--\cite{Ka2} and Takagi \cite{T}.
\subsection{Corti--Mella}\label{ssec!CM}
The typical case, and the starting point of \cite{CM}, is when $Z=Z_4\subset
\PP^4$ is a quartic hypersurface with a singularity of analytic type
$xy=z^3+t^3$. Then $Z$ is algebraically factorial, so in the Mori category.
Corti and Mella prove that the $(2,1,1,1)$ and $(1,2,1,1)$ weighted blowups
of the singular point are divisorial extractions. Each of these blowups
leads to a Sarkisov link of type~II as just described:
$$\refstepcounter{subsection}
\begin{matrix}
&& \Bl Z_4 \\
& \kern-2mm \swarrow \kern-2mm && \kern-2mm \searrow \kern-2mm \\
Z_4 &&&& (X_5 \subset\PP(1^4,2)) && \broken &
Y_{3,4} \subset\PP(1^4,2^2) \\
&&&& \kern-1cm\hbox{cont'g $\PP^2:(x_0=y=0)$}\kern-1cm &&&
\hbox{general element}
\end{matrix}
\label{eq!CM}
\eqno{(\thesubsection)}
$$
I have only described the easy part of Corti and Mella's argument,
constructing the link (\ref{eq!CM}) as an application of a fairly trivial
piece of algebra. The hard part of their work is to show that $Z_4$ and
$Y_{3,4}$ are a {\em birationally rigid pair\/}: that is, any Mori fibre
space birational to them is biregular to $Z_4$ or $Y_{3,4}$. This is the
problem of excluding links to any other Mori fibre spaces. For this, in
addition to the technology of \cite{CPR} and Corti \cite{Co2}, they need
to prove that the {\em only} extremal extractions from the singular point
$xy=z^3+t^3$ are the $(2,1,1,1)$ and $(1,2,1,1)$ weighted blowups.
\section{Varieties and graded rings, $\Proj R$, Hilbert series}
Everyone knows the correspondence
$$\refstepcounter{subsection}
X=\Proj R,\Oh_X(1) \longleftrightarrow
R=\directsum_{n\ge0} H^0(X,\Oh_X(n))
\label{eq!Proj}
\eqno{(\thesubsection)}
$$
between projective varieties and graded rings. See for example [EGA2] or
[Hartshorne], Chapter II. With the exception of Section~\ref{sec!MoriA}, I
assume that the ring is $\NN$-graded, that is, $R_n\ne0$ only for $n\ge0$,
and $R_0=k$ (the ground field $k=\CC$). The ring $R$ is almost never
generated in degree~1, so that $\Oh_X(k)$ is not necessarily determined by
$\Oh_X(1)$, and I should really specify $(X,\directsum_{k\in\ZZ}\Oh_X(k))$
on the l-h.s.\ of (\ref{eq!Proj}); for our purposes it is usually enough
to take $\Oh_X(k)=\Oh_X(kD)$ for some ample Weil divisor.
\subsection{Tutorial on Hilbert series}\label{ssec!tutH}
One of the standard applications of graded rings is when the
Hilbert\footnote{The letter $P$ stands for Poincar\'e. The technique is so
called because it was first used systematically by Cayley and Sylvester in
the context of invariant theory. I recently asked a couple of math
historians where to find Cayley and Sylvester's treatment, and I am
indebted to them for the handy tip: read their collected works in the
library.} series $P(t)=\sum P_n t^n$ is known, where $P_n=\dim R_n$
(typically, by the Riemann--Roch formula), and we can use it to guess a
plausible form of $R$ by generators and equations, and hence a plausible
model of $X$ as a variety in a w.p.s.\ with those generators and defining
equations.
\begin{exa}
$X$ is a surface of general type with invariants $p_g=h^0(K_X)$,
$q=h^1(\Oh_X)$ and $K^2$. I assume that $q=0$, so that $X$ is regular;
using Kodaira vanishing, this implies that $H^1(X,nK_X)=0$ for all $n$, so
that the graded ring $R(X,K_X)$ is Gorenstein by \cite{GW}. Then by
Riemann--Roch
\[
P_n=\begin{cases}
1 \\
p_g \\
p_g+1+\binom{n}{2} K^2 & \hbox{for } n\ge2.
\end{cases}
\]
The Hilbert series $P(t)=\sum P_nt^n$ is thus
\[
P(t)=1+p_gt+(p_g+1+K^2)t^2+\cdots+\Bigl(p_g+\binom{n}{2} K^2\Bigr)t^n+\cdots
\]
I calculate $(1-t)P(t)$ by long multiplication; this amounts simply to
differencing the coefficients of the power series:
\[
(1-t)P(t)=1+(p_g-1)t+(1+K^2)t^2+\cdots+nK^2t^n+\cdots
\]
Again multiply by $1-t$:
\[
(1-t)^2P(t)=1+(p_g-2)t+(K^2-p_g+2)t^2+\cdots+K^2t^n+\cdots,
\]
and again, to get
$$\refstepcounter{subsection}
(1-t)^3P(t)=1+(p_g-3)t+(K^2-2p_g+4)t^2+(p_g-3)t^3+t^4.
\label{eq!Hi}
\eqno{(\thesubsection)}
$$
Notice that the polynomial is symmetric (``Gorenstein symmetry''), and
the sum of the coefficients is $K^2=\deg X$.
An important case is when $p_g\ge3$ and $|K_X|$ is free; then there are
elements $x_1,x_2,x_3\in H^0(K_X)$ that form a regular sequence for
$R(X,K_X)$, and (\ref{eq!Hi}) is the Hilbert function of the Artinian
quotient ring $R(X,K_X)/(x_1,x_2,x_3)$. In particular, all the
coefficients of (\ref{eq!Hi}) are $\ge0$. However, (\ref{eq!Hi}) holds
without any assumption on $|K_X|$, for example, even if $p_g=0$.
\end{exa}
I gave the above treatment of Hilbert series in a very simple case to
illustrate the method, but there are similar formulas and methods much
more generally. There is already, for example, quite a lot of experience
of working with Hilbert series on surfaces with quotient singularities or
3-folds with canonical singularities; compare Alt{\i}nok \cite{A1} or
Kawakita \cite{Ka}--\cite{Ka2}.
\begin{exa} In Reid \cite{R}, I considered the algebraic surface $X$ with
$p_g=3$, $q=0$ and $K^2=4$ arising as the universal cover of a $\ZZ/4$
Godeaux surface. Write $R(X,K_X)$ for the canonical ring of $X$. Its
multiplied out Hilbert polynomial\footnote{I apologise for this
unconventional use of terminology. {\em Hilbert polynomial} traditionally
means the polynomial $P_{\sF}(n)=\chi(X,\sF(n))$, which coincides with
$h^0(X,\sF(n))$ after all the cohomology has died out, when $n\gg0$. Here
I am using {\em multiplied out Hilbert polynomial} for the numerator of
the Hilbert series $P(t)=\sum P(n)t^n$ after a denominator
$\prod(1-t^{a_i})$ has been chosen, corresponding to a choice of
generators $(x_1,\dots,x_n)$. Maybe it would be better to say {\em Hilbert
numerator}, or {\em Cayley--Sylvester polynomial.}\endgraf
In most cases of interest for w.p.s., $\Oh(1)$ is not a line bundle, so
$P_{\sF}(n)$ is usually not a poly\-nomial, but one of a choice of
polynomials depending on $n$ modulo the index.} is
\begin{align*}
(1-t)^3P(t)&=1+(p_g-3)t+(K^2-2p_g+4)t^2+(p_g-3)t^3+t^4 \\
&=1+2t^2+t^4.
\end{align*}
Thus the ring needs 3 generators $x_1,x_2,x_3$ in degree~1, and 2
generators $y_1,y_3$ in degree~2 (at least). Putting in these generators
gives
\[
(1-t)^3(1-t^2)^2P(t)=1-2t^4+t^8=(1-t^4)^2.
\]
We note that this coincides with the multiplied up Hilbert polynomial of a
c.i.\ of two hypersurfaces\footnote{This is a basic exercise. [Hint: expand
$\prod\frac{1}{1-x_i}$ as the sum of all monomials in $k[x_1,x_2,\dots]$,
each with coefficient 1. Substitute $x_i\mapsto t^{a_i}$, where $\wt
x=a_i$ to prove that the Hilbert series of the weighted polynomial ring is
$\prod\frac{1}{1-t^{a_i}}$. Cutting by a regular element of degree $d$
multiplies by $(1-t^d)$, so a weighted c.i.\ has Hilbert series
$\frac{\prod(1-t^{d^j})}{\prod(1-t^{a_i})}$.] For more practice, do the
\cite{Homework}.} of degree~4:
\[
k[x_1,x_2,x_3,y_1,y_3]/(f_0,f_2).
\]
Thus a plausible model for $R(X,K_X)$ is $X=X_{4,4}\subset\PP(1^3,2^2)$.
One sees that a suitable choice of the two relations makes $X$
nonsingular, and setting
\[
x_i\mapsto\ep^i x_i\hbox{ for $i=1,2,3$ \quad and}
\quad y_i \mapsto \ep^i y_i \hbox{ for $i=1,3$}
\]
defines a fixed point free action of $\ZZ/4$ on $X$, where $\ep=\exp(2\pi
i/4)$ is a primitive 4th root of 1. In \cite{R}, I showed that every
$\ZZ/4$ Godeaux surface is obtained in this way by dividing a surface
$X=X_{4,4}\subset\PP(1^3,2^2)$ by this group action.
\end{exa}
\begin{rem}
I conclude this brief tutorial on Hilbert series with the relation between
the multiplied out Hilbert polynomial $\prod_{i=0}^n(1-t^{a_i})P_R(t)=Q(t)$
and the free resolution of the graded ring $R=R(X,\Oh_X(1))$ over the
polynomial ring $A=k[x_0,\dots,x_n]$. The generators $x_i$ are always
chosen so that $R$ is a finite module over $A$. Geometrically, this means
that the $x_i$ have no common zeros on $X$ and define a finite morphism
$\pi\colon X\to\Xbar\subset\PP(a_0,\dots,a_n)$. Then $\pi_*\Oh_X$ is a
sheaf on $\PP$ or on the image $\Xbar$ whose Serre module is the ring
$R=\directsum H^0(\Oh_X(n))$. I write the sheaf $\pi_*\Oh_X$ even when I
mean the ring $R$. (As explained in \cite{PR}, 2.4, the rigorous algebraic
treatment works via the coherent Lefschetz principle with the vertex of
the affine cone over $X$, that is, $R$ localised at the ``irrelevant''
maximal ideal, but I don't want to spend time on this.) By the Hilbert
syzygies theorem, there exists a finite free resolution
$$\refstepcounter{subsection}
0\ot\pi_*\Oh_X\ot\sL_0\ot\sL_1\ot\cdots\ot\sL_m\ot0,
\label{eq!resol}
\eqno{(\thesubsection)}
$$
where each $\sL_i$ is a free graded module, that is,
$\sL_i=\directsum\Oh_\PP(-b_{i,j})$. Here $\sL_0=\Oh_{\PP}$ if and only if
$X=\Xbar$ is embedded as a projectively normal subvariety, that is,
$k[x_0,\dots,x_n]\onto R(X,\Oh_X(1))$. Each homomorphism
$\sL_{i+1}\to\sL_i$ is a matrix whose entries are homogeneous of degrees
$b_{i+1,j}-b_{i,k}$, so that the homomorphism can be considered to be
homogeneous of degree 0. Then
\[
Q(t)=\prod_{i=0}^n(1-t^{a_i})P_R(t)=\sum (-1)^i t^{b_{i,j}}.
\]
In other words, each direct summand $\Oh_\PP(-b_{i,j})$ contributes a term
$t^{b_{i,j}}$, with the generators of $\Oh_X$ (that is, $\sL_0$) counting
positively, the relations $\sL_1$ negatively, the first syzygies positively,
second syzygies negatively and so on. Unfortunately, the polynomial
expression $Q(t)$ does not entirely determine the shape of the resolution
(\ref{eq!resol}). For example, a positive term may mean a new generator, or
a first syzygy between the relations, etc. See the sidestep in
Example~\ref{exa!Go3} for a typical instance.
The really useful thing is {\em Gorenstein symmetry\/}. If $R$ is Gorenstein,
the resolution (\ref{eq!resol}) has length equal to the codimension $m=c$.
Moreover, $\sL_c\iso(\sL_0)^\vee\tensor\Oh_\PP(-k)$, where $k$ is the
{\em adjunction number}, that is,
\[
\om_X=\om_{\PP^n}\tensor\Oh_X(k)=\Oh_X\Bigl(k-\sum a_i\Bigr),
\]
and $\sL_{c-i}\iso\sL_i^\vee\tensor\Oh(-k)$. In particular, the polynomial
$Q(t)$ is symmetric: $t^m$ and $(-1)^ct^{k-m}$ appear with the same
coefficient. In writing out $Q(t)$, I usually indicate the final term
$(-1)^ct^k$, but only write out the terms up to the centre of Gorenstein
symmetry, say something like $1-t^3-3t^4+12t^4-\cdots-t^9$.
\end{rem}
\begin{exa} If $X\subset\PP(a_0,\dots,a_n)$ is a $5\times5$ Pfaffian then
\[
\prod_{i=0}^n(1-t^{a_i})P_X(t)=
1-\sum_{i=1}^5 t_i^{b_i}+\sum_{i=1}^5 t_i^{k-b_i}-t^k
\]
where $b_l=\deg\Pf_{l}$. For example, if $X$ is a K3 surface in weighted
$\PP^5$ then $K_X=0$, so that $k=\sum a_i$, and the entries in the
skew matrix are $k-b_i-b_j$. Thus the Pfaffian
$\Pf_{l}=\Pf_{ij.i'j'}$ (where $\{l,i,j,i',j'\}=\{1,2,3,4,5\}$) has degree
\[
b_l=2k-b_i-b_j-b_{i'}-b_{j'},
\]
and hence $\sum_{i=1}^5 b_i=2k=2\sum_{i=1}^6 a_i$.
\end{exa}
\section{Constructing Fano 3-folds by unprojection, first examples}
\label{sec!cod2to3}
\begin{exa}\label{exa!6+1/2}
Given $\Pi:(x_1=x_2=x_3=0)\subset\PP^5$, construct a c.i.\
$X_{2,3}\subset\PP^5$ containing $\Pi$ and as general as possible
(preferably nonsingular, but see below). Suppose that
$$\refstepcounter{subsection}
X_{2,3} : \begin{pmatrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \end{pmatrix}
\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = 0,
\label{eq!2x3}
\eqno{(\thesubsection)}
$$
with $\deg a_i=2,\deg b_i=1$.
To contract $\Pi$ to a point, I construct a function (homogeneous form) with
pole on $\Pi$. There is a clever way of doing this (see
Section~\ref{sec!Unproj1}), but I want to start by explaining a stupid way.
If I view (\ref{eq!2x3}) as 2 linear equations in 3 variables, they have a
unique solution up to proportionality
\[
x_1 \sim a_2b_3-a_3b_2, \quad\hbox{etc.,}
\]
by Cramer's rule. This suggests setting $yx_i=A_i$, with $A_i$ the
$2\times 2$ minors of (\ref{eq!2x3}), so that
\[
y=A_i/x_i \quad\hbox{for $i=1,2,3$}
\]
gives the required rational homogeneous form of degree~2 with ideal of
denominators $(x_1,x_2,x_3)$, the ideal of $\Pi$.
Adjoining $y$ with the new equations $yx_i=A_i$ gives rise to a new variety
$Y\subset\PP(1^6,2)$ defined by the 5 Pfaffians of
$$\refstepcounter{subsection}
\begin{pmatrix}
y&a_1&a_2&a_3 \\
& b_1&b_2&b_3 \\
&& x_3&-x_2 \\
&&& x_1
\end{pmatrix}
\quad\hbox{of degrees}\quad
\begin{pmatrix}
2&2&2&2 \\
&1&1&1 \\
&&1&1 \\
&&&1
\end{pmatrix}.
\label{eq!yx1}
\eqno{(\thesubsection)}
$$
\subsection{Notation} I write
\[
M=
\begin{pmatrix}
m_{12}&m_{13}&m_{14}&m_{15} \\
& m_{23}&m_{24}&m_{25} \\
&&m_{34}&m_{35} \\
&&&m_{45}
\end{pmatrix}
\]
for a skew $5\times5$ matrix. That is, I omit the diagonal terms (which are
zero) and the $m_{ji}=-m_{ij}$ with $ik_D.
\]
\begin{theorem}[\cite{KM}, \cite{PR}] \label{th!KM}
There is a rational section $s$ of\/ $\Oh_X(k_X-k_D)$ with pole on $D$
that defines a rational map
\[
X\broken Y\subset\PP^n[s]=\Proj k[x_0,\dots,x_n,s]
\]
taking $D$ to $P_s=(0:\cdots:0:1)$. Moreover, $Y$ is again projectively
Gorenstein.
\end{theorem}
\paragraph{Sketch proof} The dualising sheaf of $D$ is given by the
adjunction formula:
\[
\om_D=\sExt^1_{\Oh_X}(\Oh_D,\om_X).
\]
Here the $\sExt$ is calculated by applying the derived functor of $\sHom$
to the short exact sequence $\sI_D\to\Oh_X\to\Oh_D$. Clearly
$\sHom(\Oh_D,\om_X)=0$ and $\sExt^1(\Oh_X,\om_X)=0$ because $\Oh_X$ is
locally projective, so that we obtain the exact sequence
\[
0\to\om_X\to\sHom(\sI_D,\om_X)\to\om_D\to0,
\]
where the last map is the Poincar\'e residue map: if $D$ is a Weil divisor,
it can be written in the vulgar form $\Oh_X(K+D)\to\Oh_D(K_D)$. Now since
$\om_D=\Oh_D(k_D)$, we can twist back to obtain
\[
\sHom(\sI_D,\Oh_X(k_X-k_D))\to\Oh_D\to0.
\]
Since also $H^1(\om_X(i))=0$ for all $i$ by the projectively Gorenstein
assumption, we deduce that there exists an element
\[
s\in\sHom(\sI_D,\Oh_X(k_X-k_D))
\]
that has residue $1\in\Oh_D$. Thus $s$ has divisor of poles exactly $D$. It
is our unprojection variable; it is the same thing as the elements $s$
calculated in an ad hoc way in Sections~\ref{sec!Bx-Ay}--\ref{sec!cod2to3},
but here it is derived in a systematic way from Grothendieck duality,
without any direct calculation.
See \cite{PR} for the proof that $Y$ is projectively Gorenstein. Note that
if we write $\sI_N(k_X-k_D)=s(\sI_D)\subset\Oh_X(k_X-k_D)$ then $N$ is the
divisor of zeros of $s$. Under $X\broken Y$, $D$ is contracted to a point,
and $N$ maps isomorphically to the hypersurface section $s=0$ of $Y$. The
point of the proof in \cite{PR} is that the isomorphism
$s\colon\sI_{D,X}\iso\sI_{N,X}(k_X-k_D)$ (as ideals in the Gorenstein
scheme $X$) implies that $D$ is projectively Gorenstein if and only if $N$
is, and then $Y$ is projectively Gorenstein because its hypersurface
section $s=0$ is isomorphic to $N$, hence projectively Gorenstein. \qed
\par\medskip
\begin{rem}\label{rem!Dneg}
When contracting a divisor $D$ in a normal variety $X$, it is traditional
to assume that $\Oh_D(-D)$ is positive in some sense; here I express this
as the comparison $k_X>k_D$ between $\om_X$ and $\om_D$. This comes to the
same thing for a Cartier divisor $D$ by the adjunction formula, but is much
more powerful in general. We do not need to assume that $X$ is normal (or
even reduced), or that $D$ is even a Weil divisor (see \cite{PR},
Example~2.2).
The standard proof of Castelnuovo's contractibility criterion consists of
persuading a line bundle $\sL$ to be very ample outside $D$ but trivial on
$D$, and to have a section that restricts to the generator $1\in\Oh_D$.
This involves nonsingularity, intersection numbers and cohomology
vanishing.
The construction of Theorem~\ref{th!KM} works instead by finding a section
\[
s\in\Hom(\sI_D,\om_X\tensor\Oh_X(-k_D))
\]
whose residue on $D$ generates $\om_D\tensor\Oh_X(-k_D)$. There are no
considerations of nonsingularity, intersection numbers or cohomology
vanishing, just direct use of the projectively Gorenstein assumption on
$X$. The extra power, as so often in my experience, comes from using the
raw form of Grothendieck--Serre duality, without trying to interpret
$\om_X$ and $\om_D$ in terms of differentials $\Om^n_X$ or line bundles
$\Oh_X(K_X+D)$ as we used to do in centuries past with nonsingular
varieties. See Section~\ref{sec!Unproj2} for generalisations.
\end{rem}
\section{Applications to Fano 3-folds}
Iano-Fletcher \cite{Fl} lists the K3s and anticanonical Fano varieties whose
graded rings are hypersurfaces or codimension~2 c.i.s. There are the ``famous
95'' families of hypersurfaces, and 84 (respectively 85) families of
codimension~2 c.i.s. The odd one out here is the remarkable codimension~2
Fano 3-fold
\[
X_{12,14}\subset\PP(2,3,4,5,6,7)
\]
that does not correspond to a family of K3s, because $H^0(-K_X)=0$.
Examples~\ref{exa!6+1/2} and~\ref{exa!2+3x1/2} above are typical cases of
unprojection from codimension~2 c.i.\ to codimension~3 Pfaffian. Several
dozen more can be found by choosing a codimension~2 c.i.\ from \cite{Fl},
16.7, Table~6 containing a suitable plane $\PP(a_1,a_2,a_3)$ as a divisor.
Alt{\i}nok \cite{A} lists the K3s whose graded rings are codimension~3
Pfaffians (69 families) and codimension~4 rings (115 confirmed families, and
another 23 plausible candidates that still require more work). Many of her
cases of codimension~4 K3s extend to Fano 3-folds (with some effort). All
but a handful of her codimension~3 and confirmed codimension~4 cases are
obtained as Type~I unprojections.
\begin{rem}\label{rem!dbase}
Both Fletcher's and Alt{\i}nok's lists contain an implicit generality
assumption that exclude, for example, the monogonal and hyperelliptic
degenerations such as $X_{2,6}\subset\PP(1^3,2,3)$ and
$X_{2,4}\subset\PP(1^4,2)$. It would be interesting to plug this gap;
linear systems on K3s are well behaved with a small number of exceptions
that are themselves clear-cut dichotomies, so there should only be a
couple of dozen new cases.
While not so interesting in themselves, these huge lists of Fano 3-folds
are now acquiring some importance, and we search them repeatedly to
discover regular patterns (for example, codimension~2 c.i.s that contain a
plane $\PP(1,a_1,a_2)$ and have only terminal singularities, generalising
Examples~\ref{exa!6+1/2}--\ref{exa!2+3x1/2}), and then to find the first
few cases where that pattern breaks down. Compare Alt{\i}nok's
Example~\ref{exa!Se1}, which looks like a weighted $5\times5$ Pfaffian on
the basis of its Hilbert polynomial, but fails one little test; all of this
can easily be automated. It is a really worthwhile project to make a
computer database containing all the known information about the lists in
searchable form -- at present, it might take several hours' search and
calculation to find, say, a codimension~4 example
$X\subset\PP(1^2,2^4,a_7,a_8)$ having $7\times\half$ and some singularity
of index $\ge5$, with multiplied out Hilbert polynomial starting in
$1-3t^2-4t^5+\cdots$, and expensive taste in cigars. A working first
version of this database, programmed by Gavin Brown but based largely on
Alt{\i}nok's thesis \cite{A}, will be included in the next export of
Magma \cite{Ma} in early summer 2001.
\end{rem}
\subsection{Takagi's lists}\label{ssec!Tk}
In his Tokyo thesis \cite{T}, Takagi Hiromichi gives a systematic
treatment of Fano 3-folds with singularities of Gorenstein index~2 and
$h^0(-K_X)=g+2$ with genus $g\ge2$. This is a major achievement,
comparable to the work of Fano, Iskovskikh and others over several decades
in the nonsingular case. To simplify, assume that the only singularities
are quotient singularities $\half$\,. Takagi's lists include several cases
of anticanonical 3-folds $X$ embedded in $\PP^7(1^a,2^b)$ with $a+b=8$ as
projectively Gorenstein codimension~4 sub\-varieties. Consider in
particular the following numerical types:
\[
\renewcommand{\arraystretch}{1.6}
\begin{array}{|c|c|c|l|}
\hline
\hbox{genus}& \hbox{singularities} & \hbox{embedding} & \hbox{Hilbert
polynomial} \\
\hline
\hline
g=4 & 2\times\half & X \subset\PP(1^6,2^2) &
1-t^2-7t^3+7t^4-\cdots+t^9\\
\hline
g=3 & 3\times\half & X \subset\PP(1^5,2^3) &
1-6t^3-t^4+12t^5-\cdots+t^{10}\\
\hline
g=2 & 4\times\half & X \subset\PP(1^4,2^4) &
1-3t^3-6t^4+8t^5+\cdots+t^{11}\\
\hline
\end{array}
\]
Takagi gives a rigorous geometric treatment of every variety with these
invariants. His work is in terms of Mori theory, so that, for example, he
constructs varieties, their blowups and morphisms between them using the
MMP, rather than by calculating graded rings. However, he pointed out to
me that each of these numerical cases in codimension~4 gives rise to 2
different types of variety, and made the beautiful and almost certainly
correct prediction that these probably correspond to the two families of
unprojection treated by Papadakis \cite{P}, that we call {\em Tom} and
{\em Jerry\/}. I verify this here in the $g=4$ cases (the $g=2$ and $g=3$
cases would make fun exercises). In this case, projecting from either of
the $\half$ singularities gives a Fano 3-fold $X\subset\PP(1^6,2)$ with
multiplied out Hilbert polynomial $1-t^2-4t^3+4t^4+t^5-t^7$, which is a
$5\times5$ Pfaffian given by a matrix of degrees
$$\refstepcounter{subsection}
\begin{pmatrix}
2&2&2&2 \\
&1&1&1 \\
&&1&1 \\
&&&1
\end{pmatrix}.
\label{eq!22}
\eqno{(\thesubsection)}
$$
This family of 3-folds $X$ was constructed in Example~\ref{exa!6+1/2} by
unprojecting a plane, but now I require that it {\em contains another
plane $\Pi$.} By choosing coordinates, I assume $\Pi=\PP^2(x_4,x_5,x_6)$,
defined by $x_1=x_2=x_3=y_1=0$.
The point of Tom and Jerry is this:
\begin{quote}
\em There are two quite different ways of putting $\Pi$ inside $X$.
\end{quote}
\begin{exa}[Takagi, No.~4.4]\label{ex!Tk}
The first method is to assume that the bottom right $4\times4$ block of the
$5\times5$ matrix consists of linear combinations of the given regular
sequence $x_1,x_2,x_3,y_1$:
$$\refstepcounter{subsection}
M=
\left(
\begin{array}{c@{\enspace}c@{\enspace}ccc}
x_4 && x_5 & x_6 & p \\[2pt]
\cline{2-5}
&\vline&x_1&x_2&y_1 \\
&\vline&&x_3&ax_2 \\
&\vline&&&bx_1
\end{array}
\right), \quad\hbox{with $\deg p=2$, $\deg a,b=1$.}
\label{eq!1tom}
\eqno{(\thesubsection)}
$$
The Pfaffians of $M$ then clearly belong to the ideal generated by
$x_1,x_2,x_3,y_1$. One sees in this case that (\ref{eq!1tom}) is the {\em
general} solution: $x_1,x_2,x_3$ and $y$ must appear with unit coefficients
for reasons of degree, and any other terms can be eliminated by row and
column operations. (The terms that survive cannot be eliminated:
$m_{24}=x_2$ and $m_{35}=ax_2$ are ``Pfaffian partners'', with no row or
column in common, and the same for $m_{23}=x_1$ and $m_{45}=bx_1$.)
This is the setup for a {\em Tom unprojection\/}. Theorem~\ref{th!KM}
asserts that there exists an unprojection variable $y_2$, a rational
homogeneous form of degree 2 with $\Pi$ as its divisor of poles; however,
it does not say how to construct $y_2$. This problem is solved for the
general Tom unprojection in Papadakis' thesis \cite{P}. Here the answer
specialises to
\begin{align*}
y_2x_1&=ax_4x_6-px_5, \\
y_2x_2&=bx_4x_5-px_6, \\
y_2x_3&=bx_5^2+ax_6^2, \\
y_2y_1&=abx_4^2-p^2.
\end{align*}
\end{exa}
\begin{rem} \label{rem!xsym}
In this case, the whole set of 9 equations can be given as the $4\times4$
Pfaffians of the following $6\times6$ extrasymmetric matrix
\[
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
x_4&x_5&x_6&p&y_2 \\
&x_1&x_2&y_1&p \\
&&x_3&ax_2&ax_6 \\
&&&bx_1&bx_5 \\
&&&&abx_4
\end{pmatrix}.
\]
The unprojection variable $y_2$ goes in the top right-hand corner, from
whence it multiplies the $4\times4$ block containing the regular sequence
$x_1,x_2,x_3,y_1$. The matrix is symmetric about the antidiagonal, except
that the $356$ triangle of entries $m_{35},m_{36},m_{56}$ is multiplied by
$a$ and the $456$ triangle by $b$. Of its 15 Pfaffians, the last 6 are just
repetitions or simple multiples of the first 9.
The mechanism in geometry is that the Segre embedding
$\PP^2\times\PP^2\subset\PP^8$ is a (nongeneral) linear section of
$\Grass(2,6)\subset\PP^{14}$; it is a Schubert cycle, the lines of $\PP^5$
meeting two copies of $\PP^2$ spanning $\PP^5$. Thus $\PP^2\times\PP^2$ is
defined by the Pfaffians of a $6\times6$ (nongeneral) skew matrix. In algebra,
if $N$ is a generic $3\times3$ matrix, and we write $N=A+B$ with $A$ symmetric
and $B$ skew, then the $2\times2$ minors of $N$ generate the same ideal as
the $4\times4$ Pfaffians of the skew matrix $\left(\begin{smallmatrix} B&A
\\ -A&B \end{smallmatrix}\right)$. Multiplying a triangle such as the bottom
right triangle $456$ by an indeterminate is a flat deformation.
An extrasymmetric matrix of this type appears fairly often with Tom
unprojections. To the best of my knowledge, it appeared first in Duncan
Dicks' thesis \cite{D} (see also \cite{R1}). However, the general Tom
unprojection treated in Papadakis \cite{P} is more general than this
$6\times6$ extrasymmetric format, so don't waste too much time looking for
the matrix if it does not want to come out. (Compare the end of
Example~\ref{exa!Go3}, which just fails to have an extrasymmetric format.)
\end{rem}
\begin{prop}
For general $a,b,p$, the variety $X$ defined by the Pfaffians of
(\ref{eq!1tom}) is the midpoint of a link
\[
\PP^2 \lbroken X \broken Y,
\]
where $X\broken Y$ is the unprojection discussed above that contracts
$\Pi$ to a point of $Y\subset\PP(1^6,2^2)$, and $X\broken\PP^2$ is a
conic bundle defined by the linear system $|{-K_X-\Pi}|$, or equivalently,
the ratio $x_1:x_2:x_3$.
\end{prop}
In other words, the Fano 3-fold $Y$ and its link $Y\broken\PP^2$ are in
Takagi \cite{T}, Case~4.4. I omit the proof. The main point to note is simply
that the Pfaffian equations
\begin{align*}
\Pf_3:\quad & x_6y_1 = bx_1x_4+x_2p \\
\Pf_4:\quad & x_5y_1 = ax_2x_4+x_1p
\end{align*}
imply that $y_1$ vanishes only once on $\Pi$ so $y_1\notin H^0(-2K_X-2\Pi)$.
Thus the ring $R(X,-K_X-\Pi)$ is the polynomial ring $k[x_1,x_2,x_3]$.
\begin{exa}[Takagi, No.~1.1]\label{exa!Tk1}
The other way of imposing the plane $\Pi$ on $X$ is to assume that the
$5\times5$ matrix $M$ has first two rows with all entries in the ideal
$(x_1,x_2,x_3,y_1)$; the general solution with degrees (\ref{eq!22}) is
$$\refstepcounter{subsection}
M=
\left(
\begin{array}{cccc}
y_1&a_1&a_2&a_3 \\
&x_1&x_2&x_3 \\[2pt]
\cline{2-4}
&&x_6&-x_5 \\
&&&x_4
\end{array}
\right) \quad \hbox{with} \quad
(a_1,a_2,a_3)=(x_1,x_2,x_3)A,
\label{eq!1jer}
\eqno{(\thesubsection)}
$$
where $A$ is a $3\times3$ matrix with linear entries. In other words,
$a_1,a_2,a_3$ are linear combinations of $x_1,x_2,x_3$ with coefficients of
degree~1. Clearly all the Pfaffians of $M$ belong to $(x_1,x_2,x_3,y_1)$,
so that Theorem~\ref{th!KM} again implies that there exists an unprojection
variable $y_2$ with poles along $\Pi$.
This is the setup for a {\em Jerry unprojection\/}. The equations involving
$y_2$ are treated in Papadakis \cite{P}; in general they are much more
complicated than those for Tom, but they simplify considerably in the
present case. The two Pfaffians of (\ref{eq!1jer}) not involving $y_1$ are
bilinear in $x_1,x_2,x_3$ and $x_4,x_5,x_6$:
\[
\Pf_{23.45}=
(x_1,x_2,x_3)\begin{pmatrix} x_4 \\ x_5 \\ x_6 \end{pmatrix} = 0, \quad
\hbox{and} \quad
\Pf_{13.45}=
(x_1,x_2,x_3)A\begin{pmatrix} x_4 \\ x_5 \\ x_6 \end{pmatrix} = 0.
\]
The equations (\ref{eq!1jer}) can be obtained from these two linear
equations in $(x_4,x_5,x_6)$ by solving by Cramer's rule, with
unprojection variable $y_1$ as constant of proportionality (that is,
$y_1x_4=a_2x_3-a_3x_2$, etc.). On the other hand, I can view them also as
two linear equations for 3 unknowns $(x_1,x_2,x_3)$, and solve them with
$y_2$ as constant of proportionality. As usual, this can be written as a
$5\times5$ Pfaffian:
\[
\left(
\begin{array}{cc@{\enspace}c@{\enspace}cc}
x_3&-x_2&\vline& x_4&a'_1 \\
&x_1&\vline& x_5&a'_2 \\
&&\vline& x_6&a'_3 \\
&&&& y_2
\end{array}
\right),
\quad\hbox{where} \quad
\begin{pmatrix} a'_1 \\ a'_2 \\ a'_3 \end{pmatrix}=
A\begin{pmatrix} x_4 \\ x_5 \\ x_6 \end{pmatrix}.
\]
The equation for $y_1y_2$ turns out to be
\[
y_1y_2=(x_1,x_2,x_3)A^\dag\begin{pmatrix} x_4 \\ x_5 \\ x_6 \end{pmatrix},
\]
where $A^\dag$ is the adjoint matrix of $A$.
In this case, the Pfaffian equations say that $y_1\cdot(x_4,x_5,x_6)$ are
quadratics in $x_1,x_2,x_3$, so that $y_1\in H^0(-2K_X-2\Pi)$. Thus
the ring $R(X,-K_X-\Pi)$ is the graded ring $k[x_1,x_2,x_3,y_1]$, and
$X$ is the midpoint of a link
\[
Z \lbroken X \broken Y
\]
where $Z=\PP(1^3,2)$ is the Veronese cone. Thus $Y$ is Takagi's Case~1.1.
\end{exa}
\begin{exa}
A more general Jerry unprojection (but still not the most general, see
Papadakis \cite{P}) comes from the Pfaffian form:
$$\refstepcounter{subsection}
M=
\left(
\begin{array}{cccc}
x&a_1&a_2&a_3 \\
&b_1&b_2&b_3 \\[2pt]
\cline{2-4}
&&z_3&-z_2 \\
&&&z_1
\end{array}
\right), \quad\hbox{where}\quad
\begin{aligned}
(a_1,a_2,a_3)&=(y_1,y_2,y_3)A, \\
(b_1,b_2,b_3)&=(y_1,y_2,y_3)B, \\
\end{aligned}
\label{eq!2jer}
\eqno{(\thesubsection)}
$$
with $A,B$ generic $3\times3$ matrixes. This defines a codimension~3
Gorenstein variety containing the codimension~4 c.i.\ $(x,y_1,y_2,y_3)$.
The same bilinear trick as in Example~\ref{exa!Tk1} puts the unprojection
variable $t$ into a set of Pfaffian equations
\[
\left(
\begin{array}{cc@{\enspace}c@{\enspace}cc}
y_3&-y_2&\vline& b'_1&a'_1 \\
&y_1&\vline& b'_2&a'_2 \\
&&\vline& b'_3&a'_3 \\
&&&& t
\end{array}
\right),
\ \hbox{where}\
\begin{pmatrix} a'_1 \\ a'_2 \\ a'_3 \end{pmatrix}=
A\begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix}
\ \hbox{and}\
\begin{pmatrix} b'_1 \\ b'_2 \\ b'_3 \end{pmatrix}=
B\begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix}.
\]
The ``long equation'' for $xt$ turns out to be
\[
xt=(y_1,y_2,y_3)N(A,B)\begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix}
\]
where $N(A,B)$ is a biquadratic expression\footnote{Papadakis has
calculated this more accurately, obtaining:
\[
xt=\sum(\pm1) y_{i_1}C_{i_2,j_2}D_{i_3,j_3}z_{j_1}
\quad\hbox{summed over $\{i_1,i_2,i_3\}$, $\{j_1, j_2,j_3\}=\{1,2,3\}$,}
\]
where $C=\bigwedge^2A$ and $D=\bigwedge^2B$. Compare (\ref{eq!long}).}
in the entries of $A,B$, and is a moderately horrible mess (although
presumably a covariant of the two bilinear forms).
These equations define a flat deformation of the cone over the Segre embedding
of $\PP^1\times\PP^1\times\PP^1$. To see this, I write the equations of the
latter in terms of a little $2\times2$ cube labelled with the variables
\[
\renewcommand{\arraycolsep}{.25em}
\newcommand{\myline}{\frac{\qquad}{\qquad}}
\newcommand{\myslash}{\smallsetminus}
\newcommand{\mydown}{\Big|}
\renewcommand{\arraystretch}{1.2}
\begin{matrix} \\[-40pt]
y_1 & \myline & z_3 \\
\mydown && \mydown \\
z_2 & \myline & t
\end{matrix}
\renewcommand{\arraystretch}{1.35}
\kern-47pt
\begin{matrix} \\[-20pt]
\myslash & \qquad\, & \myslash \\
&& \\
\myslash & \qquad\, & \myslash
\end{matrix}
\renewcommand{\arraystretch}{1.2}
\kern-47pt
\begin{matrix} \\[0pt]
x & \myline & y_2 \\
\mydown && \mydown \\
y_3 & \myline & z_1
\end{matrix}
\]
Then the equations of $\PP^1\times\PP^1\times\PP^1$ are
\begin{gather*}
xz_i=y_jy_k, \quad ty_i=z_jz_k \quad\hbox{for $\{i,j,k\}=\{1,2,3\}$,} \\
\hbox{and}\quad xt=y_iz_i \quad\hbox{for $i=1,2,3$.}
\end{gather*}
Projecting from $t$ gives 5 equations in the Pfaffian form
\[
\begin{pmatrix}
x & y_1 & y_2 & 0 \\
& 0 & y_2 & y_3 \\
&& z_3 & z_2 \\
&&& z_1
\end{pmatrix},
\]
which is a specialisation of (\ref{eq!2jer}).
\end{exa}
\section{Applications to surfaces of general type}\label{sec!Go}
\begin{exa} \label{exa!Go3}
In Reid \cite{R}, I calculated the canonical ring of the universal cover
$Y$ of a $\ZZ/3$ Godeaux surface. This is a regular surface with $p_g=2$,
$K^2=3$, so that, using the Hilbert series as explained in
\ref{ssec!tutH}, you see that its canonical ring $R(Y,K_Y)$ needs at least
\[
\begin{array}{l}
\hbox{2 generators $x_1,x_2$ in degree 1,} \\
\hbox{3 generators $y_0,y_1,y_2$ in degree 2, and} \\
\hbox{2 generators $z_1,z_2$ in degree 3.}
\end{array}
\]
Then
\begin{align*}
(1-t)^2(1-t^2)^3(1-t^3)^2P(t) = 1-3t^4-3t^5&-3t^6 \\
&+2t^6+6t^7+\cdots+t^{15}
\end{align*}
(by Gorenstein symmetry, $t^{15-k}$ appears together with $t^k$). The curious
sidestep $-3t^6+2t^6$ in this expression is explained as follows: in
constructing a plausible model, we expect (or can prove, see \cite{R}) that
$|2K_Y|$ is free, and so $R(Y,K_Y)$ is a finite module over the polynomial
ring $A=k[x_1,x_2,y_0,y_1,y_2]$, generated by $1,z_1,z_2$. Therefore there
must be at least 3 equations in degree 6, expressing $z_1^2,z_1z_2,z_2^2$ in
terms of this basis.
The same ring can be obtained much more simply as a Tom unprojection. Rather
amazingly, it is then a deformation of the graded ring over the Segre
embedding of $\PP^2\times\PP^2$. For this, I start from the equations of
$\PP^2\times\PP^2$ in a slightly idiosyncratic form
\[
\rank \begin{pmatrix}
x_0&y_2&z_2 \\
z_1&x_1&y_0 \\
y_1&z_0&x_2
\end{pmatrix} \le1, \quad\hbox{that is,} \quad
\begin{aligned}
x_iz_i&=y_jy_k \\
y_iz_i&=x_jx_k \\
z_jz_k&=x_iy_i
\end{aligned} \quad\hbox{for $\{i,j,k\}=\{1,2,3\}$.}
\]
The first step is to make these equations weighted homogeneous with
$\wt x_i,\linebreak[2]y_i,\linebreak[2]z_i=1,2,3$. For this, introduce a
new variable $S$ with $\wt S=3$, and modify the equations to
$$\refstepcounter{subsection} \label{eq!Seg}
\begin{aligned}
x_iz_i&=y_jy_k, \\
y_iz_i&=Sx_jx_k, \\
z_jz_k&=Sx_iy_i.
\end{aligned}
\label{eq!S}
\eqno{(\thesubsection)}
$$
Now {\em project away from $z_0$}; in other words, separate the 9 equations
(\ref{eq!Seg}) into 4 equations linear in $z_0$, of the form
\[
x_0z_0 = \hbox{something}, \quad y_0z_0 = \cdots, \quad
z_0z_1 = \cdots, \quad z_0z_2 = \cdots,
\]
and 5 equations not involving $z_0$. It is easy to mount the latter as the
Pfaffians of the $5\times5$ skew matrix:
\[
\begin{pmatrix}
x_1&x_2&y_1&-y_2 \\
&y_0&z_2&0 \\
&&0&z_1 \\
&&&Sx_0
\end{pmatrix}
\]
I now vary the entries in the bottom right $4\times4$ block to make them
into general linear combinations of $x_0,y_0,z_1,z_2$:
$$\refstepcounter{subsection}
M_0=
\left(
\begin{array}{c@{\enspace}c@{\enspace}ccc}
x_1&&x_2&y_1&-y_2 \\[2pt]
\cline{2-5}
&\vline&y_0&z_2&r_1x_0 \\
&\vline&&r_2x_0&z_1 \\
&\vline&&&Sx_0-r_0y_0
\end{array}
\right), \quad\hbox{with $\wt r_i=2$.}
\label{eq!M_0}
\eqno{(\thesubsection)}
$$
This is the data for a Tom unprojection, as in Example~\ref{ex!Tk}. The
Pfaffians of $M_0$ are clearly contained in the ideal generated by the
regular sequence $x_0,y_0,z_1,z_2$, so they define a codimension~3 Gorenstein
variety $X$ in the ambient affine space with coordinates
$x_i,y_i,z_i,\linebreak[2]S,\linebreak[2]r_i$ such that $X$ contains the
codimension~4 c.i.\ $D:(x_0=y_0=z_1=z_2=0)$. This means that we can
unproject $D$ in $X$ by Theorem~\ref{th!KM}. As before, the explicit
equations of the unprojection can be read from Papadakis' thesis:
$$\refstepcounter{subsection}
\begin{aligned}
x_0z_0&=y_1y_2+r_0x_1x_2, \\
y_0z_0&=Sx_1x_2-r_2x_1y_2-r_1x_2y_1, \\
z_0z_1&=Sx_2y_2+r_0r_1x_2^2-r_2y_2^2, \\
z_0z_2&=Sx_1y_1+r_0r_2x_1^2-r_1y_1^2.\\
\end{aligned}
\label{eq!tom}
\eqno{(\thesubsection)}
$$
It is easy to see that the set of 9 equations (\ref{eq!M_0}--\ref{eq!tom}) is
symmetric under permuting $\{0,1,2\}$, so that they could be written in terms
of the Pfaffians of 3 matrixes like $M_0$. You can also try to mount them as
a $6\times6$ extrasymmetric Pfaffian (compare Remark~\ref{rem!xsym}):
\[
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
x_1&x_2&-y_2&y_1 & z_0 \\
&y_0&r_1x_0&z_2& Sx_1-r_1y_1 \\
&&z_1&r_2x_0 & Sx_2-r_2y_2 \\
&&&Sx_0-r_0y_0 & r_0r_1x_2 \\
&&&& r_0r_2x_1
\end{pmatrix}
\]
This matrix just fails to give the full set of equations: it only gives
the equation for $x_0z_0$ multiplied by $S,r_1,r_2$.
Note that these equations define a flat deformation of the affine cone over
the Segre variety, because they specialise to it on setting $S=1$ and
$r_i=0$. My surfaces $Y$ from \cite{R} are obtained by setting
\[
x_0+x_1+x_2=z_0+z_1+z_2=0,
\]
and $r_i=\hbox{quadratic}$, $S=\hbox{cubic}$ expressions in $x_i,y_i$, and
the $\ZZ/3$ action by cyclic permutation of $(0,1,2)$. There is a little
cyclotomic change of coordinates to go from the eigencoordinates of \cite{R}
to the cyclic permutation coordinates here. The treatment of this example
originated in the observation that the equations written out in \cite{R},
pp.~86--87 as
$$\refstepcounter{subsection}
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.2}
\begin{array}{lllllll}
R_0\quad & x_2z_1 & + & x_1z_2 & = & y_1y_2-y_0^2&+\cdots \\
R_1 &&& x_2z_2 & = & y_0y_1-y_2^2&+\cdots \\
R_2 & x_1z_1 &&& = & y_0y_2-y_1^2&+\cdots \\
S_0 & y_2z_1 &+& y_1z_2 & = & (x_1x_2-x_0^2)s&+\cdots \\
S_1 &&& y_2z_2 & = & (x_0x_1-x_2^2)s&+\cdots \\
S_2 & y_1z_1 &&& = & (x_0x_2-x_1^2)s&+\cdots
\end{array}
\label{eq!Tok}
\eqno{(\thesubsection)}
$$
take the much nicer form (\ref{eq!S}) if you replace them by their
cyclotomic combinations $R_0+\ep R_1+\ep^2R_2$ and $S_0+\ep S_1+\ep^2S_2$
(taken over the 3 roots of $\ep^3=1$), and change coordinates to
$x_0+\ep x_1+\ep^2x_2$, etc.
\end{exa}
\begin{exa} \label{exa!Ci}
Takagi's list of Fano 3-folds includes
\[
\renewcommand{\arraystretch}{1.2}
\begin{array}{ll}
\hbox{2 codim 4 families 2.2 and 3.3} &
\hbox{$X\subset \PP(1^4,2^4)$ of degree $2+4\times\half=4$,} \\
\hbox{3 codim 5 families 2.3, 3.4, 5.1} &
\hbox{$X\subset \PP(1^4,2^5)$ of degree $2+5\times\half=9/2$,} \\
\hbox{1 codim 6 family 2.4} &
\hbox{$X\subset \PP(1^4,2^6)$ of degree $2+6\times\half=5$.}
\end{array}
\]
There are almost certainly rather simple unprojection constructions for each
of these varieties. They also have sections $S\in|{-2K_X}|$ that are
canonical surfaces with $p_g=4$ and $K_S^2=8,9,10$.
\end{exa}
\begin{prob}\label{prob!it}
Canonical surfaces with invariants in this range have been studied by
Ciliberto \cite{Ci} and Catanese \cite{Ca}--\cite{Ca2} from the point of
view of generic or ``Italian'' projection discussed in \ref{ssec!proj},
(ii). These examples are interesting test cases to compare the methods
and results of Italian versus Gorenstein projection. Thus the treatment
of Example~\ref{exa!Go3} by Gorenstein projection can be compared to the
original treatment of \cite{R}, which is a kind of Italian projection: it
treats the canonical ring $R(X,K_X)$ as a module over the subring
$k[x_1,x_2,y_0,y_1,y_2]$ generated by 1 and the $z_i$, with the equations
(\ref{eq!Tok}) as defining relations. As another example, it seems clear
that Ciliberto's surfaces with $p_g=4$, $K^2=8$ must either form two
families, sections of Takagi's Cases~2.2 and~3.3, or only one family which
is a section of both. I would very much like to know which of these holds.
Compare the del Pezzo surface of degree 6, which is a linear section of
{\em both} $\PP^1\times\PP^1\times\PP^1$ and $\PP^2\times\PP^2$. This
bifurcation of cases in going from surfaces to 3-folds seems to be at the
heart of the codimension~4 Gorenstein problem. Compare
Problem~\ref{prob!inters}.
\end{prob}
\section{Tom and Jerry: who are they?}\label{sec!T&J}
Many examples of codimension~4 Gorenstein rings with a $9\times16$
resolution seem to relate to $\PP^2\times\PP^2$ or
$\PP^1\times\PP^1\times\PP^1$, although it seems hard at present to say
anything precise and general along these lines. Tom unprojections often
relate to $\PP^2\times\PP^2$ and Jerry unprojections to
$\PP^1\times\PP^1\times\PP^1$, but the short names have the advantage that
they do not imply any immodest claim concerning our current understanding
of Gorenstein codimension~4.
\begin{prob}\label{prob!intr}
Give an intrinsic treatment of Tom and Jerry.
\end{prob}
Write $C\Grass(2,5)\subset\bigwedge^2\CC^5$ for the affine cone over
$\Grass(2,5)$, that is, the generic $5\times5$ Pfaffian variety. It is an
almost homogeneous space under $\GL(5,\CC)$, and in particular has an
action of the centre $(\CC^*)^5$, which gives many choices of gradings.
A Pfaffian subvariety $X\subset A$ in a regular local scheme $A=\Spec\Oh$
is the inverse image $X=\fie\1(C\Grass(2,5))$ of $C\Grass(2,5)$ under a
morphism $\fie\colon A\to\bigwedge^2\CC^5$. To set up unprojection data, we
want $X$ to contain a given codimension~4 c.i.\
$D:(x_1=\cdots=x_4=0)\subset A$. There is presumably no loss of generality
in taking the regular sequence $x_1,\dots,x_4\in\Oh$ as part of a regular
system of parameters of $\Oh$.
Tom and Jerry each achieve $X\supset D$ by requiring that $\fie$ take $D$
to a Schubert cell:
\begin{description}
\item[Tom] The condition on $\fie$ is that $\fie(D)$ consists of
2-dimensional subspaces containing $e_1=(1,0,0,0,0)$; or $\fie(D)\subset
e_1\wedge\CC^5$. Algebraically, the skew matrix defining $X$ has bottom
right $4\times4$ block contained in the ideal of $D$:
\[
\fie^*(a_{ij})\in (x_1,\dots,x_4) \quad\hbox{for $i,j\ge2$.}
\]
\item[Jerry] In this case, $\fie(D)$ must consist of 2-dimensional
subspaces contained in $\CC^3=\Span{e_3,e_4,e_5}\subset\CC^5$; or
$\fie(D)\subset\Grass(2,\CC^3)$. That is, two rows and columns of the
matrix are contained in the ideal of $D$:
\[
\fie^*(a_{ij})\in(x_1,\dots,x_4) \quad\hbox{for $i\le2$ or $j\le2$.}
\]
\end{description}
The point of the problem, however, is to give also a description in
intrinsic terms of the unprojected variety and its equations. Compare
Papadakis \cite{P}.
\begin{prob}
Do Tom and Jerry account for {\em every\/} set of unprojection data
$D\subset X\subset A$ where $D$ is a codimension~4 c.i.\ and $X$ is a
$5\times5$ Pfaffian?
\end{prob}
The cone $C\Grass(2,5)$ over the whole Grassmann variety does not have any
divisors to unproject, so we are going to cut it down a bit by equations
forming a regular sequence, but probably not very general, until we get an
$X$ with some interesting class group. But it is then a very strong
restriction to ask an effective divisor $D$ in $X$ to be a codimension~4
c.i.\ in the ambient space.
\begin{prob}\label{prob!cu_ex}
Can all the currently known Gorenstein codimension~4 rings with
$9\times16$ resolution be accommodated within Tom and Jerry unprojection
structures?
Alt{\i}nok's treatment of codimension~4 K3 surfaces includes 23 candidates
that cannot be obtained as Type~I unprojections from codimension~3 (see
Example~\ref{exa!Se1}). In Example~\ref{exa!Se2}, I discuss a Fano 3-fold,
also derived from Alt{\i}nok's work, that has a Type~II projection, but no
Type~I projection. However, it is quite conceivable that these cases could
be part of a bigger variety that does project nicely, by analogy with
Example~\ref{exa!Ci}.
The case that I really do not know how to do at present is Duncan Dicks'
``rolling factors format'' of Example~\ref{exa!Dicks} (see also Dicks
\cite{D} and Reid \cite{R2}, Section~5). If this can't be done, it
possibly casts doubt on the whole sentiment of Problem~\ref{prob!cu_ex}.
One unresolved issue is whether Jerry (say) is a {\em structure\/} in its
own right, or a link or {\em relation\/} between two structures.
Example~\ref{exa!Dicks} is a kind of structure (``fat
$\PP^1\times\PP^1\times\PP^1$'') that at present I don't know how to relate
to $5\times5$ Pfaffians by a Jerry unprojection.
\end{prob}
\begin{rem}
Kustin and Miller remark that the generic $(2k+1)\times(2k+1)$ Pfaffian
is an unprojection (see \cite{KM}, p.~311): you can separate the variables
into $m_{12}$ and the remaining $m_{ij}$, and view the two Pfaffians not
involving $m_{12}$ as defining a codimension~2 c.i., and those involving
$m_{12}$ linearly as unprojection equations. (See Example~\ref{exa!6+1/2}
for a $5\times5$ case.)
I want to stress that this only works as stated for a {\em sufficiently
general} matrix. In fact, the generic $(2k+1)\times(2k+1)$ Pfaffian variety
is the $(k-1)$st secant variety of $\Grass(2,2k+1)$, because a skew form of
rank $\le 2k-2$ can be written as a sum $\sum e_i\wedge f_i$ of $k-1$ forms
of rank~2. The projection eliminates the variable $m_{12}$; for it to work,
$m_{12}$ must be algebraically independent of the other $m_{ij}$.
Geometrically, the point $P_{m_{12}}=(1,0,\dots,0)$ must be in the variety
in order to act as a centre of a projection. In other words, a (possibly
nongeneric) $(2k+1)\times(2k+1)$ Pfaffian variety $X$ can only have a
projection of Type~I if it has a point of the smallest possible rank~2,
that is, a point of $\Grass(2,2k+1)$.
More generally, to see a variety as an unprojection, you must first find a
suitable centre of projection, and you may well have to put your variety in
a bigger one first before this is possible. This happened in both
Example~\ref{exa!Go3} and Example~\ref{exa!Ci}. Thus in Example~\ref{exa!Ci},
if you only consider the surface $S$, you cannot see the $\half$
singularities of the 3-fold $X$, and thus its unprojection structure. For
example, in the codimension~4 cases, the 4 new generators $y_i$ in degree~2
can be thought of as a dual basis to the $4\times\half$ singularities. If
we take $\sum_{i=1}^4 y_i=0$, we have lost all the possible centres of
Gorenstein projection.
A reasonable conclusion is that the dimension of a variety is not a very
significant invariant in these constructions, and it is a mistake to
concentrate only on curves or surfaces or 3-folds. Instead, one should work
with notions such as codimension, coindex, genus, multiplied out Hilbert
poly\-nomial, homo\-logical properties, etc., that are invariant or transform
in a simple way on taking a hyperplane section, and work for preference with
a ``key variety'' that is as fat as possible.
\end{rem}
\begin{prob}\label{prob!inters}
How do Tom and Jerry intersect? As with their celluloid namesakes, scenes
in which Tom and Jerry appear together are on the whole more interesting
than their solo performances. As mentioned at the end of
Problem~\ref{prob!it}, this includes the famous deformation theory of the
del Pezzo surface of degree~6.
\end{prob}
\subsection{Why Pfaffians?}
As I tell my students, mounting a set of half-understood equations in the
form of Pfaffians is much more fun than doing crosswords, and moreover, has
some intellectual content. Apart from personal addiction, there are several
other reasons why Pfaffians turn up throughout this kind of calculation:
\begin{enumerate}
\item They are an effective and simple way of handling syzygies. If you
have written down two or three equations, and suspect that you have probably
missed one or two more, you have to do things like $x_1f_2-x_2f_1$ to cancel
some leading term and make the combination a pure multiple of $x_3$. At the
end of it, you have $f_3$ and $f_4$ with some simple linear identities. Most
frequently, the equations themselves can be written as $2\times2$ minors or
$4\times4$ Pfaffians in a way that gives the 3-term or 4-term syzygies in an
automatic way.
As we see in Section~\ref{sec!MoriA}, there are serial unprojection
rings of arbitrary codimension determined by a representative set of
equations and syzygies given as $5\times5$ Pfaffians.
\item Most current questions on Gorenstein rings are concerned with small
codimension, meaning $3,4,5,6$, and in particular with unprojecting from
codimension~2 or 3 to codimension~4 or 5 or 6. The prominence of Pfaffians
in this study is not surprising in view of the Buchsbaum--Eisenbud theorem.
Pfaffians bigger than $5\times5$ tend not to appear in this study because
they give varieties of high coindex. The $(2k+1)\times(2k+1)$ unweighted
case already has coindex $2k-2$; for example, the simplest Gorenstein graded
ring over a surface with $7\times7$ Pfaffian structure is the canonical
surface $S_{14}\subset\PP^5$.
\item The $4\times4$ Pfaffians are the Pl\"ucker equations defining
$\Grass(2,n)$. There is a natural progression $2\times2$ minors $\to$
Pfaffians $\to$ the quadratic equations defining the codimension~5 spinor
variety $\Spin(5,10)\subset\PP^{15}$ (or orthogonal Grassmann variety,
see Mukai \cite{Mu}). Just as a $4\times4$ Pfaffian is a trinomial that
you can think of as $m_{12}m_{34}-\left|\begin{smallmatrix} m_{13} &
m_{14} \\ m_{23} & m_{24} \end{smallmatrix}\right|$, each spinor equation
is a 4-nomial that you can think of as
\[
x_1x_2-\hbox{$4\times4$ Pfaffian}.
\]
\end{enumerate}
\begin{prob}
As a Pfaffian addict, I can't wait to start on the codimension~5 rings,
where the spinor equations play a similarly prominent role. According to
Mukai \cite{Mu}, the spinor coordinates $\xi_I$ on the spinor space (=
even Clifford algebra) $\CC^{16}=\bigwedge^0(\CC^5)\oplus\bigwedge^2(\CC^5)
\oplus\bigwedge^4(\CC^5)$ are indexed by even subsets of $\{1,2,3,4,5\}$.
The equations defining the spinor variety $\Spin(5,10)\subset\PP^{15}$
are the 10 spinor equations $N_{\pm i}$, typically
\[
\begin{array}{rcl}
N_1 &=& \xi_{\phi}\xi_{2345}
-\xi_{23}\xi_{45}+\xi_{24}\xi_{35}-\xi_{25}\xi_{34} \\
N_{-1} &=& \xi_{12}\xi_{1345}
-\xi_{13}\xi_{1245}+\xi_{14}\xi_{1235}-\xi_{15}\xi_{1234}.
\end{array}
\]
$\Spin(5,10)\subset\PP^{15}$ has coindex~3 and the multi\-plied out
Hilbert polynomial $1-10t^2+16t^3-16t^5+10t^6-t^8$ (the same as for a
canonical curve of genus 7, or a nonsingular Fano 3-fold of genus 7). In
the unweighted case, as for a nonsingular Fano 3-fold, a point projection
has the wrong discrepancy (see \cite{PR}, 2.7); a Fano style projection
from a line should go to a $5\times5$ Pfaffian containing a cubic scroll,
providing the first case of a Type~III unprojection from codimension~3 to
codimension~5.
That was the unweighted form. We can find many weighted homogeneous forms,
because the affine cone $C\Spin(5,10)\subset\CC^{16}$ has an action of
$\GL(5)$ and of its centre $(\CC^*)^5$. It would be interesting to find
weighted K3s and Fanos as sections of the weighted spinor varieties. This
is the simplest structure for codimension~5 Gorenstein rings, analogous to
the codimension~4 structure $5\times5$ Pfaffian intersect a hypersurface
(think of the del Pezzo surface of degree 5). The next problem is then to
find some nice examples of links by projection from these varieties to
lower codimension.
\end{prob}
\section{Harder unprojections}\label{sec!Unproj2}
We can relax the assumption in Theorem~\ref{th!KM}: there is no special
need for the divisor $D$ to be Gorenstein in order to unproject it. The
best way to think of this is from the top: Fano and the generations
following him project nonsingular Fano \hbox{3-folds} from a line, a
conic, or project doubly from a point (compare the discussion in
\ref{ssec!proj}). From \cite{CPR} and Takagi \cite{T} onwards, we can do
more exotic projections from points or curves on Fanos in the Mori
category. The exceptional divisor is usually not projectively Gorenstein.
I discuss here two families of examples:
\begin{description}
\item[Type~II] In this case $D$ is not projectively Cohen--Macaulay,
because it is not projectively normal, but the normalisation
$\Oh_{\wD}=\Oh_D\oplus\Oh_D\cdot t$ needs only one module generator, and
moreover, $\Oh_{\wD}$ is Gorenstein. This arises in connection with the
elliptic involution of \cite{CPR}, 4.10--4.12 and~7.3, and with several
interesting codimension~4 K3s and Fano 3-folds from Alt{\i}nok's lists
\cite{A}. See Examples~\ref{sec!Key2}--\ref{exa!Se2}. It is really a
generic phenomenon for slightly nonnormal embeddings
$\PP(a_1,a_2,b)\into\PP(a_1,a_2,a_3,\dots,a_n)$ between w.p.s.s.
\item[Type~III] In this case, $D$ is projectively Cohen--Macaulay, but
$\om_D(n_0)$ is generated by {\em two} sections that define a fibre space
structure $\fie\colon D\to\PP^1$. That is, $D$ is homologically like a
cubic scroll, so Cohen--Macaulay but not Gorenstein. The typical case is
Fano's projection of a Fano 3-fold from a line, with the cubic scroll as
exceptional divisor. Corti suggested treating the inverse rational map as
a new type of unprojection, and calculated the first cases himself. See
Example~\ref{exa!Ta2.1}, where I conclude the story begun in
Example~\ref{exa!2+3x1/2} based on Takagi, Case~2.1.
\end{description}
\begin{rem}
By assigning roman numerals, I am certainly not suggesting a case division
or classification. Rather, these are certain pathologies that turn up
frequently, and that we can begin to handle alongside the Kustin and Miller
Type~I cases. Based on experience of projecting Fano \hbox{3-folds} from
different centres (and seeking the unprojection giving the left-hand side
of the corresponding link), I believe that $D$ can be really very bad from
the point of view of commutative and homological algebra. There are
certainly cases when $D$ is a badly nonnormal scroll, or when $\Oh_{\wD}$
is Gorenstein but needs many generators as an $\Oh_D$ module.
\end{rem}
\begin{prob}\label{prob!genU}
Find the best theorem of the following shape. I state the problem in the
local setup. Compare \cite{PR}, 2.4 for the translation from local to
projective.
\begin{quote} \em
$X$ is a local Gorenstein scheme and $D\subset X$ a subscheme
of pure codimension~$1$. The adjunction formula for $\om_D$ gives the usual
exact sequence
\[
0\to\om_X\to\sHom(\sI_D,\om_X)\to\om_D\to0.
\]
Identifying $\om_X=\Oh_X$ interprets elements of the $\sHom$ as rational
functions on $X$ with poles along $D$. Pick a set of generators
$s_i\in\sHom(\sI_D,\om_X)$, say with $s_0=\id\colon\sI_D\subset\Oh_X=\om_X$.
As in \cite{PR}, Lemma~1.1, assume without loss of generality that the $s_i$
are injective and have divisor of poles exactly $D$.
Define the unprojection (ring) of $D$ in $X$ by
$$\refstepcounter{subsection}
\Oh_Y=\Oh_X[s_1,\dots,s_n]/(\mathrm{relations}) \quad\hbox{and}\quad
Y=\Spec\Oh_Y.
\label{eq!locU}
\eqno{(\thesubsection)}
$$
Then under suitable (fairly mild) conditions, $Y$ is a Gorenstein scheme.
\end{quote}
It is part of the problem to say what the ideal of relations in
(\ref{eq!locU}) should be. When it turns out that $Y$ is birational to
$X$, we could just take the relations between the $s_i$ holding in the
total ring of fractions $k(X)$, but in general $Y$ may have new
components.
Maybe we should find the relations by studying a few examples; it would be
really cool if all the relations were determined by linear ones.
Conjecture~\ref{conj!cool} suggests that in some easy cases, we should
look for linear relations in the $s_i$ and certain fairly simple and
predictable quadratic relations yoked to them by Pfaffians.
\end{prob}
In the projective setup, in view of the applications, I want to assume that
$D$ is a codimension~1 subscheme of a projectively Gorenstein scheme $X$,
and that there is a threshold value $k_D\in\ZZ$ with $k_X>k_D$ for which
$\om_D(-k_D)$ is still generated by its $H^0$, but the resulting linear
system is not big, so that the morphism $\fie_{\om_D(-k_D)}$ contracts $D$
to a smaller dimensional variety.
As before, this means that the elements $s_i\in\sHom(\sI_D,\om_X(-k_D))$
whose residues generate $\om_D(-k_D)$ become homogeneous forms with poles
along $D$, and have positive degree $k_X-k_D$ under the identification
$\om_X=\Oh_X(k_X)$.
\begin{rem} As in Remark~\ref{rem!Dneg}, the assumption $k_X>k_D$ is a
negativity condition on $D\subset X$. Note the fortunate circumstance that
the $s_i$ correspond to generators of $\om_D(-k_D)=\om_{\wD}(-k_D)$, which
is good even if $D$ is not normal. A condition expressed in terms of
$\Oh_D$ would be much worse in this respect. This is another advantage of
the approach via Grothendieck--Serre duality.
In the modern view, we usually expect this kind of canonical threshold to
be a rational number. But $k_D\in\QQ$ does not seem to make sense here. (Or
could it somehow?)
\end{rem}
\subsection{Key variety for Type~II unprojections} \label{sec!Key2}
I give a generic form for Type~II unprojections, as a preparation
for the following examples. Consider the morphism
\[
\begin{array}{rcl}
\pi\colon\wD=\CC^{n+1} &\to& D \subset\CC^{2n+1} \\
(x_1,\dots,x_n,t) &\mapsto& (x_i,y_i=x_it,z=t^2)
\end{array}
\]
that folds the $t$ axis in half, identifying $\pm t$. The image $D$ is a
nonnormal toric variety, with coordinate ring the subring $k[D]\subset
k[x_1,\dots,x_n,t]$ obtained by outlawing odd pure powers of $t$. Its
equations are
$$\refstepcounter{subsection}
\rank N\le1, \quad\hbox{where}\quad N=\begin{pmatrix}
y_1&\cdots&y_n&x_1z&\cdots&x_nz \\
x_1&\cdots&x_n&y_1&\cdots&y_n
\end{pmatrix};
\label{eq!2x2n}
\eqno{(\thesubsection)}
$$
that is, the $n^2$ equations
$$\refstepcounter{subsection}
\begin{cases}
\renewcommand{\arraystretch}{1.2}
x_iy_j-x_jy_i &\hbox{for $i0$, $b_0=1$, $a_k=0$
and $b_l<0$, and the side tags form complementary continued fractions.}
\label{fig!cmp}
\end{figure}
Convexity considerations and combinatorics of concatenated continued
fractions (compare Craw and Reid \cite{CR}, Section~2) reduce us to just a
few cases. The main one is the rectangle of Figure~\ref{fig!cmp} with tags
forming {\em complementary continued fractions\/}
\begin{equation}
[a_0,a_1,\dots,a_{k-2}]=\frac{n}{q}\quad\hbox{and}\quad
[b_1,b_2,\dots,b_{l-1}]=\frac{n}{n-q}
\label{eq!ccf}
\end{equation}
for some $n$ and $q$.
This is a well known way of fixing up identities such as the second of
(\ref{eq!circum}); see for example \cite{CR}, Figure~4 or \cite{Rie}, \S3,
pp.~220--3. However, as Jan Stevens taught us, it is also a way of {\em
deconstructing} the continued fraction, successively eliminating the tag 1
and decrementing its two neighbours. For example, take $n=11$ and $q=7$;
then $\frac{11}{7}=[2,3,2,2]$ and $\frac{11}{4}=[3,4]$. Concatenating the
two continued fractions with a 1 gives $[2,3,2,\underline{2,1,4},3]$, that
deconstructs by replacing $2,1,4$ by $1,3$:
\[
\to [2,3,\underline{2,1,3},3]
\to [2,\underline{3,1,2},3] \to [2,\underline{2,1,3}]
\to [\underline{2,1,2}] \to [1,1]=0.
\]
Compare \cite{CR}, \S2. My main point is:
\begin{quote} \em
the same calculation gives successive Gorenstein projections of the toric
coordinate ring $k[V_u]$ down to a codimension~$2$ complete intersection.
\end{quote}
In fact, if $y_0$ has the tag $b_0=1$ then also $x_0,y_1$ and $u$ base
$M$. This is basically the same reason that allowed us to assume that
$a_i,b_j\ge2$ down the sides (if $a_i=1$, I can eliminate $x_i$ as a
generator), but here eliminating $y_0$ cuts down the cone $\si$, so makes
a birational change to $V_u$.
To see the effect of this change in more detail, note that because $y_0$
has the tag $b_0=1$, it occurs linearly in 3 tag equations:
\[
x_1y_0=x_0u^\al, \quad y_0y_2=y_1^{b_1} \quad\hbox{and}\quad
x_0y_1=y_0u^\be
\]
(and also in long equations $y_0x_i$ for $i\ge2$ and $y_0y_j$ for $j\ge3$,
but we do not need these). To eliminate $y_0$, I multiply the first two
equations by $u^\be$ and substitute $y_0u^\be=x_0y_1$ in each, then cancel
a power of $x_0$ or $y_1$ respectively, to get the new tag equations
\[
x_1y_1=x_0^{a_0-1}u^{\al+\be} \quad\hbox{and}\quad
x_0y_2=y_1^{b_1-1}u^\be.
\]
In words, chop off the top right-hand corner of $\si$,
\begin{figure}
\begin{picture}(150,80)(-130,0)
\put(0,75){$u^{\al+\be}$}
\put(13,65){$a_0-1$}
\put(28,58){\line(5,-2){100}}
\put(25,55){$\bullet$}
\put(28,58){\line(0,-1){60}}
\put(12,15){$a_1\,\bullet$}
\put(125,15){$\bullet\,b_1-1$}
\put(145,30){$u^{\be}$}
\put(128,17){\line(0,-1){19}}
\end{picture}
\caption{Projecting from $y_0$}
\label{fig!chop}
\end{figure}
giving the new Gorenstein quadrilateral cone $\si'$ with top corners given
as in Figure~\ref{fig!chop}. Note that the two monomials adjacent to the
recently executed $y_0$ have their tags decremented by~1, but inherit a
factor of $u^\be$ in their annotation.
Under the current assumption that $[a_0,a_1,\dots,a_{k-2}]$ and
$[b_1,b_2,\dots,b_{l-1}]$ are complementary continued fractions, one of
$a_0-1$ and $b_1-1$ is again equal to 1, allowing me to eliminate $x_0$ or
$y_1$ by another Gorenstein projection, and so on. The serial projection
ends with a rectangle
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{2.5}
\begin{array}{rc}
\raisebox{3.5mm}{\mbox{$u^p$}}\ d-1 & \bullet \\
0 & \bullet
\end{array}
\kern-2.05mm
\framebox[17.5mm]{\rule[-5.7mm]{0mm}{10.6mm}\qquad}
\kern-2.0mm
\begin{array}{cl}
\bullet & 0 \ \raisebox{3.5mm}{\mbox{$u^q$}} \\
\bullet & -(d-1)
\end{array}
\]
where the exponents $p,q$ of $u$ are linear combinations of $\al$ and $\be$
determined by cumulatively multiplying the annotations. This rectangle
represents the codimension~2 complete intersection
\[
x_{k-1}y_l=u^q, \quad x_ky_{l-1}=x_{k-1}^{d-1}u^p.
\]
\subsection{A pair of long rectangles}
\begin{exan}\label{ex!ki4}
I illustrate in a simple codimension~4 case how to deform a Gorenstein
toric variety by projection and unprojection to obtain the $\GG_m$ cover
of a Mori flip. The quadrilateral cone of monomials and its tag equations
are as follows:
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{2.0}
\begin{array}{rcl}
\raisebox{3.5mm}{\mbox{$u^\al$}}\ 1 & \bullet &
\raisebox{-2.5mm}{\mbox{$x_0$}}\\
2 & \bullet & x_1 \\
d & \bullet & x_2 \\
\raisebox{-3.5mm}{\mbox{$u^{2\al+\be}$}}\ 0 & \bullet &
\raisebox{2.8mm}{\mbox{$x_3$}}
\end{array}
\kern-8.2mm
\framebox[22.5mm]{\rule[-14.1mm]{0mm}{28.6mm}\qquad}
\kern-8.0mm
\begin{array}{rcl}
\raisebox{-2.5mm}{\mbox{$y_0$}} & \bullet & 2 \
\raisebox{3.5mm}{\mbox{$u^\be$}} \\
\\
\\
\raisebox{2.8mm}{\mbox{$y_1$}} & \bullet & -(d-1)
\end{array}
\kern-0.5em\implies\quad
\begin{array}{rclrcl}
x_1y_0&=&x_0u^\al\quad &x_0y_1&=&y_0^2u^\be \\
x_0x_2&=&x_1^2 \\
x_1x_3&=&x_2^d \\
x_2y_1&=&u^{2\al+\be}\quad &x_3y_0&=&x_2^{d-1}u^\al \\
\end{array}
\]
for some $\al>0$ and $\be\ge0$.
\begin{remn}
The bottom right equation $x_3y_0=x_2^{d-1}u^\al$ is nonstandard: because
the tag $b_1=-(d-1)$ is negative, the regular tag equation would be
\begin{equation}
x_3y_0=y_1^{-(d-1)}u^?,
\label{eq!alcove}
\end{equation}
which is not a polynomial equation. I therefore replace $y\1_1$ by
$x_2u^?$ using the bottom left equation, getting the {\em modified tag
equation} $x_3y_0=x_2^{d-1}u^?$. This replaces the negative exponent of
the corner monomial $y_1$ with {\em a positive exponent of the monomial
$x_2$ opposite the corner\/}. We certainly pay for this substitution when
we do syzygies, although I do not know how to express this sentiment
mathematically.
Note also that (\ref{eq!alcove}), rewritten as $x_3y_0y_1^{d-1}=u^?$, is a
general feature of the monomial rectangle (\ref{fig!Gor_cone}) that may at
first sight seem somewhat unexpected: the internal monomial $u$ chooses to
live right in one corner, namely in the convex hull of $y_l,x_k,y_{l-1}$.
\end{remn}
I deform this ring by guessing the two equations at the top:
\begin{equation}
x_1y_0=x_0u^\al+t^{2\la+\mu}
\quad\hbox{and}\quad
x_0y_1=y_0^2u^\be+x_1t^{\la}
\label{eq!ki_star}
\end{equation}
where $t$ is an elephantine deformation parameter ($t=0$ will be the
anticanonical section for the $\ZZ$-grading). There is nothing very special
about the exponents of $t$: they are a priori arbitrary coprime integers
satisfying some inequalities and possibly divisibility conditions; the
point of writing them in this form is to make the bottom two equations and
the $\ZZ$-grading pretty.
Now (\ref{eq!ki_star}) is a codimension~2 c.i.\ that contains the
codimension~3 c.i.\ $(x_0,y_0,t^\la)$. It thus unprojects by
Theorem~\ref{th!KM}. The unprojection variable $x_2$ satisfies 3 new
equations given by a game of Pfaffians similar to that of
Example~\ref{exa!6+1/2}:
\[
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
y_1&-x_1&-y_0u^\be&x_2 \\
& y_0&-t^\la&-u^\al \\
&& x_0&-t^{\la+\mu} \\
&&& x_1
\end{pmatrix} \ \implies\
\left\{
\begin{array}{rcl}
x_0x_2&=&x_1^2+y_0u^\be t^{\la+\mu} \\
x_2y_0&=&x_1u^\al+y_1t^{\la+\mu} \\
x_1y_1&=&y_0u^{\al+\be}+x_2t^\la
\end{array}
\right.
\]
These 5 equations define a Gorenstein codimension~3 variety that contains
the codimension~4 c.i.\ $(x_0,x_1,y_0,t^\la)$. This again unprojects by
Theorem~\ref{th!KM}, adjoining $x_3$. In fact, it is a Jerry unprojection
(see Example~\ref{exa!Tk1}): rows and columns Nos.~3 and~4 of the Pfaffian
matrix (all entries except for the 125 triangle) are in the ideal
$(x_0,x_1,y_0,t^\la)$.
As in Example~\ref{exa!Tk1}, there is an easy trick to derive $x_3$ as a
rational function, namely elimination of $x_0$ (a projection). Notice that
$x_0$ appears linearly in 3 of our 5 Pfaffians; the two not involving
$x_0$ are the above equations
\[
x_2y_0=x_1u^\al+y_1t^{\la+\mu} \quad\hbox{and}\quad
x_1y_1=y_0u^{\al+\be}+x_2t^\la.
\]
These define a codimension~2 c.i.\ that contains two separate
codimension~3 c.i.s, namely the ideal of denominators $(x_2,y_1,u^\al)$ of
$x_0$ and the new one $(x_1,y_0,t^\la)$ that is the ideal of denominators
of $x_3$. I indulge myself in just one final round of Pfaffians:
\[
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
y_1&-x_2&-u^{\al+\be}&x_3 \\
& y_0&-t^\la&-u^\al \\
&& x_1&-y_1t^\mu \\
&&& x_2
\end{pmatrix} \ \implies\
\left\{
\begin{array}{rcl}
x_1x_3&=&x_2^2+y_1u^{\al+\be}t^\mu \\
x_3y_0&=&x_2u^\al+y_1^2t^\mu \\
x_2y_1&=&u^{2\al+\be}+x_3t^\la
\end{array}
\right.
\]
The first of these equations proves that if we make a deformation of the
ring in Example~\ref{ex!ki4}, starting from the top two equations
(\ref{eq!ki_star}) and adopting the above style of unprojection, then
necessarily $d=2$. As in Example~\ref{exa!Tk1}, we do not really have a
good way of deriving the ``long equation'' for $x_0x_3$. As far as we
know, it is not contained in a Pfaffian in any useful way. Messing around
with explicit syzygies eventually gives
\begin{equation}
x_0x_3=x_1x_2+y_0y_1u^\be t^\mu+u^{\al+\be}t^{\la+\mu}.
\label{eq!long}
\end{equation}
There is a unique way of putting a $\ZZ$-grading on this set of equations
with $\wt u=0$, $\wt t=-1$, provided that $3\mid\mu$. Namely,
\begin{align*}
\wt x_3=\la>0, & \quad \wt y_1=\mu/3>0, \quad\hbox{and} \\
\wt x_2=-\mu/3, & \quad
\wt x_1=-\la-2\mu/3, \\
& \wt x_0=-2\la-\mu, \quad
\wt y_0=-\la-\mu/3.
\end{align*}
The $\GG_m$ quotient is the flip diagram $X\searrow Y\swarrow X^+$, where
\[
X =\Proj R_- =
\left(
\begin{array}{rcl}
x_1y_0&=&x_0u^\al+t^{2\la+\mu} \\
x_0y_1&=&y_0^2u^\be+x_1t^{\la}
\end{array}
\right)
\Big/\GG_m
\]
is covered by the two affine pieces $x_0=1$ and $y_0=1$ and
\[
X^+=\Proj R_+ =
\left(
\begin{array}{rcl}
x_3y_0&=&x_2u^\al+y_1^2t^\mu \\
x_2y_1&=&u^{2\al+\be}+x_3t^\la
\end{array}
\right)
\Big/\GG_m
\]
is covered by the two affine pieces $x_3=1$ and $y_1=1$.
This is a Mori flip of Type~A, with $t=0$ the general elephant, and $u=0$
the general hyperplane section. For example, the $y_0=1$ affine piece of
the left-hand side $X$ is the hyperquotient singularity
\[
\left(
\begin{array}{rcl}
x_1&=&x_0u^\al+t^{2\la+\mu} \\
x_0y_1&=&u^\be+(x_0u^\al+t^{2\la+\mu})t^{\la}
\end{array}
\right)
\Big/\frac{1}{\wt y_0}(\wt x_0,\wt y_1,0,1).
\]
If we look up the weights, we see that this is
\[
(x_0y_1=u^\be+\cdots+t^{3r})
\big/\frac{1}{r}(-a,a,0,1), \quad\text{with $r=\la-\frac{\mu}{3}$,
$a=\frac{\mu}{3}$},
\]
which is a standard Type~A terminal singularity.
\end{exan}
\subsection{Conclusions from this example} \label{ssec!concl}
The rectangle of Example~\ref{ex!ki4} defines an affine Gorenstein toric
3-fold $V_u$; I have shown how to deform it to a 4-fold $V_{u,t}$ with a
$\GG_m$ action such that $V_{u,t}$ has isolated singularities modulo the
action and isolated fixed points. Requiring isolated fixed points means
that the corner equations contain pure powers of $t$ (possibly after a
substitution). The deformation style adopted keeps track of the powers of
\begin{figure}[ht]
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{2.0}
\begin{array}{rcl}
\raisebox{3.5mm}{\mbox{$A$}}\ 1 & \bullet \\
2 & \bullet \\
d & \bullet \\
\raisebox{-3.5mm}{\mbox{$A^2B$}}\ 0 & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-14.1mm]{0mm}{28.6mm}\qquad}
\kern-4.2mm
\begin{array}{rcl}
& \bullet & 2 \
\raisebox{3.5mm}{\mbox{$B$}} \\
\\
\\
& \bullet & -(d-1)
\end{array}
\quad\hbox{and}\quad
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{2.0}
\begin{array}{rcl}
\raisebox{3.5mm}{\mbox{$L^2M$}}\ 0 & \bullet &
\\
2 & \bullet & \\
2 & \bullet & \\
\raisebox{-3.5mm}{\mbox{$L$}}\ 1 & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-14.1mm]{0mm}{28.6mm}\qquad}
\kern-4.2mm
\begin{array}{rcl}
& \bullet & -1 \\
\\
\\
& \bullet & 2 \
\raisebox{-3.5mm}{\mbox{$M$}}
\end{array}
\]
\caption{The pair of long rectangles for Example~\ref{ex!ki4}}
\label{fig!pairKi4}
\end{figure}
$t$ by introducing the right-hand rectangle of Figure~\ref{fig!pairKi4},
having different top and bottom corner tags and annotations, but identical
torso. This imposes $d=2$ on the original rectangle.
As in Remark~\ref{rem!V}, the final expression only contains $u$ and $t$
within the tokens $u^\al,u^\be$, $t^\la,t^\mu$. Replacing
\[
u^\al\mapsto A,\quad u^\be\mapsto B,\quad
t^\la\mapsto L,\quad t^\mu\mapsto M
\]
in the equations (say, taking the top two equations to $x_1y_0=x_0A+L^2M$
and $x_0y_1=y_0^2B+x_1L$) gives an affine Gorenstein 6-fold $V_{ABLM}$ with
a regular sequence $A,B,L,M\in k[V_{A,B,L,M}]$ such that the codimension~2
sections $V_{AB}:(L=M=0)$ and $V_{LM}:(A=B=0)$ are the toric 4-folds with
respective cones of monomials documented by the long rectangles of
Figure~\ref{fig!pairKi4}. There is a 4-dimensional torus $\GG_m^4$ with a
monomial action on $V_{ABLM}$, namely, the subgroup of the diagonal group
$\GG_m^8$ acting on $\CC^8$ with coordinates $x_0,x_1,y_0,y_1,A,B,L,M$
that leaves semi-invariant the top two equations $x_1y_0=x_0A+L^2M$ and
$x_0y_1=y_0^2B+x_1L$.
\begin{remn} \label{rem!kA_1}
At the start of Example~\ref{ex!ki4}, the assumption on $\be$ was only that
$\be\ge0$. If $\be>0$ then $\PP^1(x_0:y_0)$ is contained in the elephant
and the point $P_{y_0}=(0:1)\in V_u$ is a terminal singularity of index
$(-\wt y_0)$. If $\be=0$, the top right equation contains a pure power of
$y_0$, so that $P_{y_0}\notin V_u$. In this case, $S^-\to S$ of
(\ref{eq!fl}) is an isomorphism. This is Mori's distinction between cases
$kA_2$ and $kA_1$, having respectively two and one singularities of index
$>1$ on $S$. In this case the angle at $y_0$ in the left-hand long
rectangle straightens out, so that the elephant is represented by a ``long
triangle''.
\end{remn}
\subsection{Pairs of long rectangles and serial unprojection via
pentagrams}\label{ssec!pair}
The combinatorial classification of pairs of long rectangles is solved in
Brown and Reid \cite{BR}. We obtain a number of families labelled by Roman
numerals: the case of Figure~\ref{fig!pairKi4} is currently called
$\three(1,0)$. Each pair gives rise to a {\em two-headed toric $6$-fold\/}
$V_{ABLM}$ by the serial unprojection method outlined below, with the
properties sketched in \ref{ssec!concl}. We still have some work to do
to identify our treatment with Mori's calculation \cite{M}, but the pairs
$\two(d,e,k)$ of Figure~\ref{fig!roma_two} seem to be most closely related
to it.
\begin{figure}[h]
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.4}
\begin{array}[t]{rcl}
\raisebox{3.5mm}{\mbox{$A$}}\ d & \bullet \\
e & \bullet \\
d & \bullet \\
\vdots \\
d & \bullet \\
0 & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-34mm]{0mm}{33.8mm}\qquad}
\kern-4.8mm
\renewcommand{\arraystretch}{1}
\begin{array}[t]{rcl}
& \bullet & 1 \
\raisebox{3.5mm}{\mbox{$B$}} \\
& \equiv & 2 \times (d-2) \\
& \bullet & 3 \\
& \equiv & 2 \times (e-3) \\
& \bullet & 3 \\[-4pt]
&& \vdots \\
& \equiv & 2 \times (e-2)\\
& \bullet & -(d-1) \\
\end{array}
\quad\raisebox{-17mm}{\hbox{ and }}\quad
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.4}
\begin{array}[t]{rcl}
\raisebox{3.5mm}{\mbox{$\ $}}\ 0 & \bullet \\
& \bullet \\
& \bullet \\
\vdots \\
& \bullet \\
\raisebox{-3.5mm}{\mbox{$L$}}\ e & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-34mm]{0mm}{33.8mm}\qquad}
\kern-4.8mm
\renewcommand{\arraystretch}{1}
\begin{array}[t]{rcl}
& \bullet & -(e-1) \
\raisebox{4.5mm}{\mbox{$\ $}} \\
& \equiv & \\
& \bullet & \\
& \equiv & \\
& \bullet & \\[-4pt]
&& \vdots \\
& \equiv & \\
& \bullet & 1 \raisebox{-3.5mm}{\mbox{$M$}} \\
\end{array}
\]
\caption{The pair of long rectangles $\two(d,e,k)$.}
\label{fig!roma_two}
\end{figure}
The figure illustrates the case $k$ even and $d,e\ge4$. The two
rectangles have the same torso tags (excluding tops and tails): $k$ terms
$e,d,e,d,\dots,d$ down the left and $k$ blocks $2,\dots,2,3$ of $d-2$ and
$e-2$ terms each down the right, giving rise to complementary continued
fractions
\[
[e,d,\dots]=\frac{n}{q} \quad\hbox{and}\quad
[2,\dots,2,3,2,\dots]=\frac{n}{n-q}
\]
as in (\ref{eq!ccf}).
\begin{remn}
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item The two rectangles correspond to the elephant $t=0$ and the general
section $u=0$ of a flip. Mori has proved that if the elephant is of Type~A
then the section is a cyclic quotient singularity, this is where
$V_{ABLM}$ gets its two toric heads from.
\item We would be interested to know if the $V_{ABLM}$ have already been
studied elsewhere. We hope that they have other descriptions; it is
conceivable that they are quasihomogeneous varieties for some slightly
bigger group than $\GG_m^4$, for example, something with a unipotent
radical such as two copies of
$\left(\begin{smallmatrix}\GG_m&\GG_a\\0&\GG_m\end{smallmatrix}\right)$.
All those Pfaffian equations of $V_{ABLM}$ that become binomial (toric) on
cutting to $V_{AB}$ and $V_{LM}$ might to have something to do with
extending toric varieties to quasihomogeneous varieties modelled on
$\GL(2)$.
\item It should be reasonably straightforward to extend much of the
apparatus of toric geometry to deal with the $V_{ABLM}$. Invariant
divisors, coherent cohomology, Betti cohomology and Hodge theory, derived
categories, Gromov--Witten invariants, mirror partners\dots Get on with it,
this isn't a research grant application!
\item It seems likely that the two-headed toric varieties $V_{ABLM}$ can
also act as key varieties in other contexts. For example, we expect that
they can be given nice positive gradings, and so act as key varieties for
projective varieties coming from serial unprojection constructions, as
illustrated in Examples~\ref{exa!Go3}--\ref{exa!Ci}. This could extend the
range of our artillery for attacking surfaces and 3-folds, bringing other
interesting targets within range.
\end{enumerate}
\end{remn}
\begin{exan} \label{exa!k=3}
I illustrate serial unprojection with a little workout in the case $k=2$,
and get a final fix ({\em ultimo Pfaffiano!\/}). The bullets down the left
side of the rectangle in Figure~\ref{fig!k=3} are monomials
$x_0,x_1,x_2,x_3$. To avoid going into double suffixes, I
\begin{figure}[hb]
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.4}
\begin{array}[t]{rcl}
\raisebox{3.5mm}{\mbox{$A$}}\ d & \bullet \\
e & \bullet \\
d & \bullet \\
0 & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-19.4mm]{0mm}{19.4mm}\qquad}
\kern-4.8mm
\renewcommand{\arraystretch}{1.05}
\begin{array}[t]{rcl}
& \bullet & 1 \
\raisebox{3.5mm}{\mbox{$B$}} \\
& \equiv & 2 \times (d-2) \\
& \bullet & 3 \\
& \equiv & 2 \times (e-2) \\
& \bullet & -(d-1) \\
\end{array}
\quad\raisebox{-9mm}{\hbox{ and }}\quad
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.4}
\begin{array}[t]{rcl}
\raisebox{3.5mm}{\mbox{$\ $}}\ 0 & \bullet \\
& \bullet \\
& \bullet \\
\raisebox{-3.5mm}{\mbox{$L$}}\ e & \bullet &
\end{array}
\kern-4.2mm
\framebox[16mm]{\rule[-19.4mm]{0mm}{19.4mm}\qquad}
\kern-4.8mm
\renewcommand{\arraystretch}{1.05}
\begin{array}[t]{rcl}
& \bullet & -(e-1) \
\raisebox{4.5mm}{\mbox{$\ $}} \\
& \equiv & \\
& \bullet & \\
& \equiv & \\
& \bullet & 1 \raisebox{-3.5mm}{\mbox{$M$}} \\
\end{array}
\]
\caption{The pair of long rectangles $\two(d,e,k)$.}
\label{fig!k=3}
\end{figure}
write $y_0,\dots,y_d$ and then $z_0,\dots,z_e$ for the monomials down the
right side, with an overlap of 3:
\[
y_{d-2}=z_0, \quad \hbox{$y_{d-1}=z_1$ is the monomial with tag 3, and}
\quad y_d=z_2.
\]
I start work on the right-hand rectangle (the cone of monomials for
$V_{LM}$), with the initial objective of discovering the annotations at
its top corners. The bottom right tag is a 1, so that I can project from
$z_e$ as in Figure~\ref{fig!chop} (but bottom-up this time), then
successively from $z_{e-1},\dots,z_2$. By the rules given around
Figure~\ref{fig!chop}, the new tag at $x_3$ is decremented by 1 at each
projection, and the annotation at $x_3$ inherits a factor of $M$, so that
after $e-1$ steps the tag is 1 and the annotation is $LM^{e-1}$, giving
successive tag equations
\[
x_1x_3=x_2^d,\quad x_2z_1=x_3LM^{e-1}, \quad x_3z_0=z_1^2M.
\]
The last projection of $z_2$ decrements the tag at $z_1$ from 3 to 2, and
gives $z_1$ an annotation of $M$.
Now $x_3$ can be projected: I chop it off, and its annotation $LM^{e-1}$
is multiplied into that of $x_2$ and $z_1=y_{d-1}$. The score at
half-time is:
\[
\renewcommand{\arraycolsep}{.25em}
\renewcommand{\arraystretch}{1.6}
\begin{array}[t]{rcl}
\raisebox{3.5mm}{\mbox{$\ $}}\ 0 & \bullet \\
e & \bullet \\
\raisebox{-4.5mm}{\mbox{$LM^{e-1}$}}\ d-1 & \bullet &
\end{array}
\kern-4.2mm
\framebox[14mm]{\rule[-14.2mm]{0mm}{14.2mm}\qquad}
\kern-4.8mm
\renewcommand{\arraystretch}{1.6}
\begin{array}[t]{rcl}
& \bullet & -(e-1) \
\raisebox{4.5mm}{\mbox{$\ $}} \\
& \equiv & 2 \times d-2 \\
& \bullet & 1\ \raisebox{-4.5mm}{\mbox{$LM^e$}} \\
\end{array}
\]
Next project from $y_{d-1},\dots,y_2$. At each point the annotation $LM^e$
of $y_i$ is passed on to that of $y_{i-1}$, and is multiplied into the
tag of $x_2$. Since we project $d-2$ times, the tag of $x_2$ decrements
down to 1, and its annotation multiplies up to
\[
LM^{e-1}\times(LM^{e})^{d-2} = L^{d-1}M^{de-e-1}.
\]
Finally project $x_2$, so that its annotation is passed on to $x_1$ and
multiplied into that of $y_1$ to give $L^dM^{de-1}$. This leaves us with
a rectangle representing the two equations
\[
x_1y_0=L^dM^{de-1} \quad\hbox{and}\quad x_0y_1=x_1^{e-1}L^{d-1}M^{de-e-1}
\]
at the top corners of the right-hand rectangle.
Merging the right hand side of these with those at the top corners of the
left-hand rectangle gives the top equations for the 6-fold $V_{ABLM}$:
\begin{equation}
x_1y_0=x_0^dA+L^dM^{de-1} \quad\hbox{and}\quad
x_0y_1=y_0B+x_1^{e-1}L^{d-1}M^{de-e-1}.
\label{eq!fun}
\end{equation}
Now this is where the fun really starts. Consider the series of $5\times5$
Pfaffians
\begin{multline*}
\renewcommand{\arraystretch}{1.4}
\begin{pmatrix}
x_2&-B&-x_1^{e-1}&y_i\\
&x_1&-LM^e&-x_0^{d-i}AB^{i-1}\\
&&x_0&-x_2^{i-1}L^{d-i}M^{(d-i)e-1}\\
&&&y_{i-1}
\end{pmatrix} \\[10pt]
\implies \quad
\renewcommand{\arraystretch}{1.2}
\left\{
\begin{array}{rcl}
x_1y_{i-1}&=&x_0^{d-i+1}AB^{i-1}+x_2^{i-1}L^{d-i+1}M^{(d-i+1)e-1} \\
x_0y_i&=&y_{i-1}B+x_1^{e-1}x_2^{i-1}L^{d-i}M^{(d-i)e-1} \\
x_2y_{i-1}&=&x_0^{d-i}x_1^{e-1}AB^{i-1}+y_iLM^e \\
x_1y_i&=&x_0^{d-i}AB^{i}+x_2^iL^{d-i}M^{(d-i)e-1} \\
x_0x_2&=&x_1^e+BLM^e
\end{array}
\right.
\end{multline*}
for $i=1,\dots,d-1$. When $i=1$, the first two equations are just
(\ref{eq!fun}), and the final three, linear in $x_2$, are the equations
defining the unprojection of the codimension~3 c.i.\
$(x_0,y_0,L^{d-1}M^{de-e-1})$ in the codimension~2 c.i.\ (\ref{eq!fun}).
For $i\ge2$, the matrix is easily read off the {\em pentagram} of
Figure~\ref{fig!penta}, (a). We start from the two known equations for
$x_0x_2$ and $x_1y_{i-1}$. The 5
\begin{figure}[bht]
\[
\setlength{\unitlength}{1mm}
\begin{picture}(38,40)(8,-18)
\put(6,24){\circle*{1.44}} % x_0
\put(6,12){\circle*{1.44}} % x_1
\put(6,0){\circle*{1.44}} % x_2
\put(30,12){\circle*{1.44}} % y_{i-1}
\put(30,6){\circle*{1.44}} % y_i
\put(6,12){\line(1,0){24}} % line x_1y_{i-1}
\put(6,12){\line(4,-1){24}} % line x_1y_i
\put(6,0){\line(2,1){24}}
\qbezier(6,24)(16,12)(6,0) % arc x_0x_2
\qbezier(6,24)(18,11)(30,6) % arc x_0y_i
\put(0,23.4){$x_0$}
\put(0,11.4){$x_1$}
\put(0,-0.6){$x_2$}
\put(32,11.4){$y_{i-1}$}
\put(32,5.4){$y_i$}
\put(16,-20){(a)}
\end{picture}
\begin{picture}(38,40)(0,-18)
\put(6,24){\circle*{1.44}} % x_0
\put(6,12){\circle*{1.44}} % x_1
\put(6,0){\circle*{1.44}} % x_2
\put(6,-12){\circle*{1.44}} % x_3
\put(30,6){\circle*{1.44}} % y_i
\put(6,12){\line(4,-1){24}} % line x_1y_i
\put(6,0){\line(4,1){24}} % line x_2y_i
\qbezier(6,24)(16,12)(6,0) % arc x_0x_2
\qbezier(6,12)(16,0)(6,-12) % arc x_1x_3
\qbezier(6,24)(24,6)(6,-12) % arc x_0x_3
\put(0,23.4){$x_0$}
\put(0,11.4){$x_1$}
\put(0,-0.6){$x_2$}
\put(0,-12.6){$x_3$}
\put(32,8.4){$y_{d-1}$}
\put(34,2.4){$=z_1$}
\put(16,-20){(b)}
\end{picture}
\begin{picture}(38,40)(-10,-18)
\put(6,12){\circle*{1.44}} % x_1
\put(6,0){\circle*{1.44}} % x_2
\put(6,-12){\circle*{1.44}} % x_3
\put(30,0){\circle*{1.44}} % z_i
\put(30,6){\circle*{1.44}} % y_i
\put(6,0){\line(1,0){24}} % line x_2z_i
\put(6,12){\line(2,-1){24}} % line x_1z_i
\put(6,0){\line(4,1){24}} % x_2z_{i-1}
\qbezier(6,12)(16,0)(6,-12) % arc x_1x_3
\qbezier(6,-12)(18,2)(30,6) % arc x_1x_3
\put(0,11.4){$x_1$}
\put(0,-0.6){$x_2$}
\put(0,-12.6){$x_3$}
\put(32,-.6){$z_i$}
\put(32,5.4){$z_{i-1}$}
\put(16,-20){(c)}
\end{picture}
\]
\caption{Pentagrams for $\two(2,d,e)$}
\label{fig!penta}
\end{figure}
points of the pentagram are $x_2,x_1,x_0,y_{i-1},y_i$. We write these
cyclically around the super\-diagonal and top right of the matrix, so that
adjacent vertexes do not multiply in Pfaffians, but vertexes that are two
apart do so, as in the pentagram: the new unprojection variable $y_i$ goes
in the top right, from whence it will multiply $x_0$, $x_1$ and the middle
entry $m_{24}$, but not $x_2$ or $y_{i-1}$. The remaining matrix entries
are uniquely determined by requiring that $\Pf_{12.34}$ is the known
equation for $x_0x_2$ and $\Pf_{23.45}$ the known equation for
$x_1y_{i-1}$ with the middle entry $m_{24}=LM^e$ the hcf of the terms
$BLM^e$ and $x_2^{i-1}L^{d-i+1}M^{(d-i+1)e-1}$ in those two equations
(taking a factor smaller than the hcf would lead to a nonnormal variety).
The 3 new Pfaffians determine the new unprojection variable $y_i$ as a
rational function, and one proves via Theorem~\ref{th!KM} that it defines
a Gorenstein variety with coordinate ring generated by
$x_0,x_1,x_2,y_0,\dots,y_i,A,B,L,M$.
For $1\le i\le d-2$, suppose by induction that these equations hold for
$i$; then projecting from $y_{i-1}$ deletes the first three equations,
leaving the last two as a codimension~2 c.i.\ containing $(x_0,x_1,LM^e)$.
Thus we can introduce a new unprojection variable $y_{i+1}$, with three new
relations contained in the same set of equations with $i\mapsto i+1$.
The rest is similar. At the end of the first half, the first series of
Pfaffians culminates at $i=d-1$ with the equation
\[
x_1y_{d-1}=x_0AB^{d-1}+x_2^{d-1}LM^{e-1}.
\]
At half-time, we use this together with the equation for $x_0x_2$ as input
to a $5\times5$ Pfaffian matrix corresponding to the pentagram (b),
switching to $z_1=y_{d-1}$ for the second half:
\begin{multline*}
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
x_2&-BM&-x_1^{e-1}&x_3 \\
&x_1&-LM^{e-1}&-AB^{d-1} \\
&&x_0&-x_2^{d-1} \\
&&&z_1
\end{pmatrix} \\
\implies\quad
\renewcommand{\arraystretch}{1.2}
\begin{array}{rcl}
x_0x_3&=&x_1^{e-1}x_2^{d-1}+z_1BM \\
x_1x_3&=&x_2^d+AB^dM \\
x_2z_1&=& x_1^{e-1}AB^{d-1}+x_3LM^{e-1}
\end{array}
\end{multline*}
Notice that the middle term of the matrix $m_{24}=LM^{e-1}$ has slipped
down to the hcf of two terms in the input equations $\Pf_{12.34}$ and
$\Pf_{23.45}$.
The two last equations form the input to the series of Pfaffians
corresponding to the pentagram Figure~\ref{fig!penta}, (c) that play
during the second half:
\[
\renewcommand{\arraystretch}{1.2}
\begin{pmatrix}
x_3&-AB^d&-x_2^{d-1}&z_{i+1} \\
&x_2&-M&-x_1^{e-i-1}A^iB^{d-i} \\
&&x_1&-x_3^iLM^{e-i-1} \\
&&&z_i
\end{pmatrix}
\]
This works by induction as before, and I leave it at that.
\end{exan}
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\bibitem[A1]{A1} S. Alt{\i}nok, Hilbert series and applications to graded
rings, submitted
\bibitem[B1]{B1} G. Brown, Flips in low codimension -- classification and
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\end{thebibliography}
\medskip
\noindent
Miles Reid,\\
Math Inst., Univ. of Warwick,\\
Coventry CV4 7AL, England\\
e-mail: miles@maths.warwick.ac.uk \\
web: www.maths.warwick.ac.uk/$\!\sim$miles
\end{document}
Now this is where the fun starts. First, the unprojection of the
codimension~3 c.i.\ $(x_0,y_0,L^{d-1}M^{de-e-1})$ in the codimension~2
c.i.\ (\ref{eq!fun}) is given by the $5\times5$ Pfaffians
\begin{multline*}
\renewcommand{\arraystretch}{1.4}
\begin{pmatrix}
x_1&-LM^e&-x_0^{d-1}A&x_2\\
&x_0&-L^{d-1}M^{(de-e-1}&-B\\
&&y_0&-x_1^{e-1}\\
&&&y_1
\end{pmatrix} \\[10pt]
\implies \quad
\renewcommand{\arraystretch}{1.2}
\left\{
\begin{array}{rcl}
x_2y_0&=&x_0^{d-1}x_1^{e-1}A+y_1LM^e \\
x_1y_1&=&x_0^{d-1}AB+x_2L^{d-1}M^{de-e-1} \\
x_0x_2&=&x_1^e+BLM^e
\end{array}
\right.
\end{multline*}
Here we take $L^{d-1}M^{de-e-1}$ as the hcf of the end terms in
(\ref{eq!fun}) in order to get a normal variety after unprojection.
Next, the last two equations do not involve $y_0$, and define a c.i.\
containing the codimension~3 c.i.\ $(x_0,x_1,LM^e)$. I can thus project
away from $y_0$ (eliminate it) and unproject again, introducing $y_2$ and
so on. Consider the series of $5\times5$ Pfaffians
\begin{multline*}
\renewcommand{\arraystretch}{1.4}
\begin{pmatrix}
x_2&-B&-x_1^{e-1}&y_i\\
&x_1&-LM^e&-x_0^{d-i}AB^{i-1}\\
&&x_0&-x_2^{i-1}L^{d-i}M^{(d-i)e-1}\\
&&&y_{i-1}
\end{pmatrix} \\[10pt]
\implies \quad
\renewcommand{\arraystretch}{1.2}
\left\{
\begin{array}{rcl}
x_1y_{i-1}&=&x_0^{d-i+1}AB^{i-1}+x_2^{i-1}L^{d-i+1}M^{(d-i+1)e-1} \\
x_0y_i&=&y_{i-1}B+x_1^{e-1}L^{d-i}M^{(d-i)e-1} \\
x_2y_{i-1}&=&x_0^{d-i}x_1^{e-1}AB^{i-1}+y_iLM^e \\
x_1y_i&=&x_0^{d-i}AB^{i}+x_2^iL^{d-i}M^{(d-i)e-1} \\
x_0x_2&=&x_1^e+BLM^e
\end{array}
\right.
\end{multline*}
for $i=1,\dots,d-1$. When $i=1$, the first two equations are just
(\ref{eq!fun}), and the final three, linear in $x_2$, are the equations
defining
For $1\le i\le d-2$, if we know these equations hold, project from
$y_{i-1}$ deletes the first three equations, leaving the last two as a
codimension~2 c.i.\ containing $(x_0,x_1,LM^e)$. Thus we can introduce
a new unprojection variable $y_{i+1}$, with three new relations contained
in the same set of equations with $i\mapsto i+1$.