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%% Orbifold Riemann--Roch and plurigenera
%% Anita Buckley and Miles Reid
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\begin{document}
\title{Orbifold Riemann--Roch and plurigenera}
\author{Anita Buckley and Miles Reid}
\date{}
\maketitle
\begin{abstract}
We give a general formula for the Hilbert series of a simply polarised
$n$-dimensional orbifold with isolated orbifold points. The result comes
from orbifold RR, and so ultimately from equivariant RR (the
Atiyah--Singer Lefschetz trace formula); however, the formula is
organised so that no Chern or Todd classes appear explicitly, and no
Dedekind sums. The formula reduces much of the senior author's work over
20 years to a few lines of computer algebra. \end{abstract}
For dramatic effect, we state a simple case of the theorem first, leaving
definitions and explanations for later.
\begin{theorem} \label{thm!first}
Let $\bigl(X,\,\bigoplus\Oh_X(i)\bigr)$ be a simply polarised $n$-fold
with $n\ge1$. Assume that $X$ is projectively Gorenstein with canonical
weight $k_X$ and has a basket of isolated orbifold points
$\sB=\bigl\{\rth(a_1,\dots,a_n)\bigr\}$ as its only singularities.
Then the Hilbert series $P_X(t) = \sum_{n\ge0} h^0(X,\Oh_X(n))t^n$ of
$X$ is
\begin{equation}
P_X(t)=P_I(t)+\sum_\sB P_\orb(\rth(a_1,\dots,a_n),k_X),
\end{equation}
with {\em initial term} $P_I(t)$ and {\em orbifold terms}
$P_\orb(\rth(a_1,\dots,a_n),k_X)$ characterised as follows:
\begin{itemize}
\item $P_I(t) = \frac{A(t)}{(1-t)^{n+1}}$ has denominator $(1-t)^{n+1}$,
and numerator $A(t)$ an integral Gorenstein symmetric polynomial of
degree the {\em coindex} $c=k_X+n+1$ of $X$, so that $P_I(t)$ equals
$P(t)$ up to degree $\left[\frac c2\right]$.
\item Each orbifold term $P_\orb(\rth(a_1,\dots,a_n),k_X\bigr)=
\frac{B(t)}{(1-t)^n(1-t^r)}$ has denominator $(1-t)^n(1-t^r)$, and
numerator $B(t)$ the unique integral Laurent poly\-nomial supported in
$\Bigl[ \left[\frac c2\right]+1,\left[\frac c2\right]+r-1 \Bigr]$ which
is
\[
\hbox{the inverse modulo}\quad \frac{1-t^r}{1-t}=1+t+\cdots+t^{r-1}
\quad\hbox{of}\quad \prod\frac{1-t^{a_i}}{1-t}.
\]
\end{itemize}
\end{theorem}
The initial term $P_I(t)$ determines and is determined by the first
$\left[\frac c2\right]$ plurigenera, and is $0$ if $c<0$; for example,
weighted projective space $\PP(a_0,\dots,a_n)$ with at least one
$a_i>1$. The coefficients of the orbifold terms are closely related to
Dedekind sums, but are determined by conceptually simple {\em ice cream
functions} (see Example~\ref{exa!IC}) and are given by easy computer
algebra routines.
\begin{exa}[``Ice cream on Wednesdays, Fridays and Sundays'']
\label{exa!IC} \rm \
\noindent The step function $i\mapsto[3i/7]$ is familiar. As a Hilbert
series, it gives
\begin{equation}
P(t):=\sum_{i\ge0}[3i/7]t^i=0+0t+0t^2+t^3+t^4+2t^5+2t^6+3t^7+\cdots
\end{equation}
This series takes the closed form
\begin{equation}
P(t)=\frac{t^3+t^5+t^7}{(1-t)(1-t^7)}.
\end{equation}
In fact, since $[3i/7]$ increases cumulatively by 1 at $i=3,5,7$ mod~7,
it follows that $(1-t)P(t)$ is the sum of $t^i$ taken over the jumps
\begin{equation}
(1-t)P(t) = \sum_{i\ge0}t^i = t^3+t^5+t^7+t^{10}+ \cdots
\end{equation}
repeating periodically with period~7. Multiplying by $1-t^7$ cuts this
down to the first week's ice cream ration.
The numerator $t^3+t^5+t^7$ is the inverse of
$(1-t^5)/(1-t)=1+t+t^2+t^3+t^4$ modulo
$(1-t^7)/(1-t)=1+t+t^2+t^3+t^4+t^5+t^6$ (here 5 is the inverse of 3
mod~7). {\bf Proof:} the product $(1+t+t^2+t^3+t^4)(t^3+t^5+t^7)$
consists of 15 terms, distributed equitably among the 7 powers of $t$
modulo~$t^7$, except that we hit $t^7$ three times, so that
$(1+t+t^2+t^3+t^4)(t^3+t^5+t^7)=$
\begin{align*}
\setcounter{MaxMatrixCols}{20}
\renewcommand{\arraycolsep}{.125em}
& \begin{matrix}
1&+&t&+&t^2&+&t^3&+&t^4 \\
&&&&&+&t^3&+&t^4&+&t^5&+&t^6&+&t^7 \\
&&&&&&&&&+&t^5&+&t^6&+&t^7&+&t^8&+&t^9 \end{matrix} \\
&\equiv 3+2t+2t^2+\cdots+2t^6\ \hbox{mod~$1-t^7$} \\
&\equiv 1\ \hbox{mod~$1+t+t^2+t^3+t^4+t^5+t^6$.}
\end{align*}
There are several other meaningful expressions for $P(t)$: working
modulo $1+t+t^2+t^3+t^4+t^5+t^6$, one can view the bounty $t^3+t^5+t^7$
as famine $-t-t^2-t^4-t^6$ (``no ice cream on Mondays, Tuesdays,
Thursdays and Saturdays''), that is,
\begin{equation}
P(t)=\frac{t^3+t^5+t^7}{(1-t)(1-t^7)}
=\frac1{(1-t)^2}+\frac{-t-t^2-t^4-t^6}{(1-t^7)}.
\end{equation}
Either of these functions can be shifted up or down, e.g., to give
\begin{equation}
\frac{t^{-4}+t^{-2}+1}{(1-t)(1-t^7)} \quad\hbox{or}\quad
\frac{-t^{-1}-t-t^2-t^4}{(1-t)(1-t^7)}, \quad\hbox{etc.,}
\end{equation}
``ice cream from the week (or day) before term''.
Or the macroeconomic view is that $[3i/7]$ is the linear function $3i/7$
with periodic corrections, giving
\begin{equation}
P(t)=\frac37\times\frac1{(1-t)^2}
+\frac{-\frac37t-\frac67t^2-\frac27t^3-\frac57t^4-\frac17t^5-\frac47t^6}{1-t^7}\,.
\end{equation}
Notice that the day we lose the smallest residue $\frac17$ in small change
is Friday, corresponding to $3\times5=1$ mod 7.
\end{exa}
The coefficients here are Dedekind sums. We will see that the general
$P_\orb$ and general Dedekind sums are obtained by minor variations on
this simple calculation.
\begin{exa}\rm The general curve $C_{15}\subset \PP(1,5,7)_{x,y,z}$ is
quasismooth with an orbifold point of type $\frac17(5)$ at the point
$P_z=(0,0,1)$, and having orbinate $y$ (or $\frac{y}{z^{5/7}}$) there. It
is an elliptic curve polarised by $A=\frac37(P_z)$. It is projectively
Gorenstein with orbifold canonical class
\[
K_{C,\mathrm{orb}}=\frac67P_z=(15-1-5-7)A=2A
\]
and $k_C=2$, so coindex $c=4$. Its Hilbert series is
\[
\frac{1-t^{15}}{(1-t)(1-t^5)(1-t^7)}
= \frac{1-t-t^3+t^4}{(1-t)^2}+\frac{t^3+t^5+t^7}{(1-t)(1-t^7)}
\]
There is a lot to learn from this simple example; please check it all
for yourself.
\end{exa}
\section{Background material}
We start with some easy and well known generalities. Let $R$ be an
$\N$-graded ring, assumed to be finitely generated in strictly positive
degrees over $\C$. A finite $R$-module $M$ may have negative terms, but
finitely generated implies $M_l=0$ for $l>-l_0$, and each $M_l$ is a
finite dimensional $\C$-vector space. As usual, define the {\em Hilbert
series} of $M$ by
\[
P_l(M)=\dim_\C M_l \quad\hbox{and}\quad
P_M = \sum_{l\in\Z} P_l(M)t^l.
\]
\begin{prop} Suppose that $R=k[x_1,\dots,x_N]/I_R$ where the generators
$x_i$ are homo\-geneous of weight $a_i$. Then $P_M$ is of the form
\[
P_M=\frac{\hbox{integral Laurent polynomial in $t$}}
{\prod_{i=1}^N(1-t^{a_i})}.
\]
\end{prop}
\paragraph{Proof} Induction on $N$. Let $x=x_N$ and $a=a_N$; consider
the multiplication $x\colon M\to M$. Its kernel $K$ and cokernel $N$
are graded modules fitting into exact sequences
\begin{equation}
0 \to K_{l-a} \to M_{l-a} \xrightarrow{\,x\,} M_l \to N_l \to 0
\label{eq!mx}
\end{equation}
Since $x$ acts by 0 on $K$ and $N$, they are modules over a smaller
ring, so their Hilbert series are of the required form by induction. On
the other hand, \eqref{eq!mx} gives $P_l(M)-P_{l-a}(M)=P_l(N)-P_{l-a}K$
for each $l$, and summing gives
\[
(1-t^a)P_M(t)=P_N(t)-t^aP_K(t). \QED
\]
\section{Introduction} \subsection{Terminology}\label{ssec!term}
\begin{defn}\label{defn!Porb} \rm
A {\em simply polarised orbifold with isolated orbifold points\/} is a
variety $X$ polarised by a sheaf of graded algebras $\bigoplus\Oh_X(i)$
satisfying:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $X$ is a projective $n$-fold over a field $k$ (e.g., $k=\C$), and
$\Oh_X(m)$ is an ample invertible sheaf for some $m>0$;
\item $X$ has at most isolated orbifold singularities
$\rth(a_1,\dots,a_n)$, and locally at each point, each $\Oh(i)$ is
isomorphic to the $i$th eigensheaf of the $\mu_r$ action.
\end{enumerate}
Note that {\em simply} polarised includes two conditions: the grading is
by $\N$ (as opposed to lattice polarised or graded by a more complicated
semigroup); also $\Oh_X(1)$ generates the local divisor class
$\Cl(X,P)$, that is, provides a polarisation of the singularity.
\end{defn}
Assume in this introduction that orbifold behaviour occurs at isolated
points in codimension~$\ge2$, so $n=\dim X\ge2$. The methods also apply
to orbifold behaviour in codimension~1 or~0 after some elementary stacky
preliminaries; see \ref{ssec!curve} for orbifold curves. Then the
$\Oh(i)$ are divisorial sheaves, and $\Oh(i)=\Oh_X(iA)$ for an ample
Weil divisor $A$. In this case the {\em graded structure sheaf}
$\bigoplus\Oh_X(i)$ is specified by $A$ or $\Oh(1)=\Oh(A)$.
Under the assumptions of Definition~\ref{defn!Porb}, the graded ring
\begin{equation}
R(X)=R\Bigl(X,\bigoplus\Oh_X(i)\Bigr)=\bigoplus_{i\ge0} H^0(X,\Oh_X(i))
\end{equation}
is a finitely generated $k$-algebra, of the form
$R(X)=k[x_0,\dots,x_N]/I_X$ with weighted generators $x_i\in
H^0(X,\Oh_X(w_i))$ and a weighted homogeneous ideal $I_X$. The affine
variety $\sC_X=\Spec X$ is the {\em weighted cone} over $X$; the grading
induces an action of the multiplicative group $\G_m$ on $R(X)$ and
$\sC_X$ that defines the quotient $X=\Proj R(X)=(\sC_X\setminus0)/\G_m$.
Under our assumptions, $\sC_X$ is nonsingular outside the origin, and the
orbifold behaviour of $X,\bigoplus\Oh_X(i)$ comes from isolated orbits
with cyclic isotropy subgroups $\mu_r\subset\G_m$. The generators $x_i$
give $\sC_X\subset\aff^{N+1}$ and $X\subset\PP(w_0,\dots,w_N)$, where
$\aff^{N+1}$ is affine space with coordinates $x_0,\dots,x_N$ and
$\PP^N(w_0,\dots,w_N)$ is weighted projective space (wps or w$\PP^N$)
with homogeneous coordinates $x_i$ of weight $w_i$.
\begin{defn} \rm Write $P_i(R)=\dim_k R_i$ for the dimension of the
$i$th graded piece of a finitely generated graded ring
$R=\bigoplus_{i\ge0} R_i$; by abuse, we call
$P_i(X)=P_i(R(X))=h^0(X,\Oh(i))$ the $i$th {\em plurigenus} of $X$. The
{\em Hilbert series} of $R(X)$ or of $X$ is the formal power series
\begin{equation}
P_X(t)=P_{R(X)}(t)=\sum_{i\ge0} P_i(R)t^i.
\end{equation}
It is known to be a rational function $\Num(t)/\prod(1-t^{w_j})$ with
denominator corresponding to the generators of $R$. The main point of
this paper is that the generating function $P_X(t)$ is often simpler
than the individual $P_i(X)$. Our problem is to calculate $P_X(t)$ under
extra conditions. \end{defn}
\begin{defn}\label{defn!pGor} \rm We say that $X,\bigoplus\Oh_X(i)$ is
{\em projectively Gorenstein} if $R(X)$ is a Gorenstein graded ring.
This is equivalent (compare \cite{GW} and \cite{W}) to the following
cohomological conditions:
\begin{itemize}
\item $H^j(X,\Oh_X(i))=0$ for all $j$ with $0k_X$, it makes
sense to divide the formal sum up as a formal power series $P_X(t)$ in
positive powers of $t$ plus $(-1)^nt^{k_X}P_X(1/t)$, which is $t^{k_X}$
times a power series in negative powers of $t$. On the other hand,
multiplying the formal sum formally by $(1-t)^{n+1}$ gives zero, since
$\chi(\Oh_X(i))$ is a polynomial of degree $n$ in $i$. This proves
(\ref{eq!fn}).
(\ref{eq!fn}) implies that the numerator of $P(X)$ is a symmetric
polynomial, and completes the proof. \QED
\begin{rmk}\label{rmk!initP} \rm For low values of $c$, the numerator
$\Num(t)$ in (\ref{eq!Num}) is
\begin{equation}
\begin{array}{|c|l|}
\hline \hbox{coindex} & \Num(t) \\ \hline \hline c=0 & 1 \\ c=1 & 1+t \\
c=2 & 1+dt+t^2 \\ c=3 & 1+(g-2)t+(g-2)t^2+t^3 \\ c=4 &
1+at+bt^2+at^3+t^4 \\ c=5 & 1+at+bt^2+bt^3+at^4+t^5 \\ \hline
\end{array}
\end{equation}
This is the Hilbert series converse of Remark~\ref{rmk!init}. The form
of these poly\-nomials is familiar: for example, a regular surface of
general type has $c=4$, with $a=p_g-3$ and $b=K^2-(2p_g-4)$. The formula
itself works perfectly well even if $p_g=0$, so $a=-3$. The sum of the
coefficients, that is $\Num(1)$, equals the degree of the polarised
variety $X$.
Our convention is to take the first $[c/2]$ coefficients of $\Num(t)$ as
the basic global invariants of $X$. One effect is that we study Hilbert
series in terms of plurigenera themselves; relating the initial
plurigenera to the topological invariants of $X$ (the Todd classes, the
terms in $\int\ch(iA)\Td(X)$ of Hirzebruch RR) becomes a secondary
issue.
The classic case is when $R(X)$ has a regular sequence $x_0,\dots,x_n$
in degree~1; geometrically, this means that $|\Oh_X(1)|$ is a free
linear system. Then $R(X)$ is a free graded module over the polynomial
ring $k[x_0,\dots,x_n]$, and its generators map one-to-one to a
$k$-vector space basis of the Artinian quotient ring
$R(X)/(x_0,\dots,x_n)$; the Hilbert numerator of $R(X)$ is the Hilbert
series of this Artinian quotient. Passing to the numerator of
$P_X(t)=\frac{\Num(X)}{(1-t)^{n+1}}$ has the effect of normalising $X$
to dimension~$-1$. \end{rmk}
\subsection{Temporary workaround in the curve case} \label{ssec!curve}
Our treatment so far has avoided the full graded structure sheaf of
$X,\Oh_X(i)$ by assuming that $X$ only has orbifold behaviour in
codimension $\ge2$, so that $X$ is normal and $\Oh_X(i)=\Oh_X(iA)$ with
$A$ a $\Q$-Cartier Weil divisor, determined by $\Oh(1)=\Oh(A)$. However,
orbifold behaviour in codimension~1 and~0 is natural and simple, and
can't be avoided if we want a treatment of orbifolds that includes
induction by restriction to smaller strata. Then the simple
$\Oh(i)=\Oh(iA)$ device does not work: it is not true that the sheaf
$\Oh_X(i)$ is determined by $\Oh_X(1)$, because the multiplication maps
$\Oh_C(i)\tensor\Oh_C(j)\to\Oh_C(i+j)$ are no longer isomorphisms in
codimension~1.
The way around this in the curve case is simple and well
known:\footnote{Other drafts of the same remark:
The point can be viewed in terms of stacks: the space is only the
underlying space $|X|$ (or coarse moduli space) for the bigger structure
$X,\bigoplus\Oh(i)$. In particular, a general orbifold has a graded
dualising sheaf $\om_X(i)=\Oh_X(k_X-i)$.
The material of \ref{ssec!term} uses the traditional ``well-formed''
device of higher dimensional geometry that allows one to avoid
mentioning stacks when the orbifold behaviour is in codimension~$\ge2$:
work with a Weil $\Q$-Cartier divisor $A$ and set $\Oh_X(i)=\Oh_X(iA)$.
\begin{rmk}[Orbifold in codim 0 and 1] \rm When $n=1$, the orbifold
points in codimension~1 require extra care: the notation
$\Oh(i)=\Oh_X(iA)$ only makes sense after introducing a $\Q$-Weil
divisor $A=\sum \frac{a_j}{r_j}E_j$, as in Demazure \cite{De}, and
$\Oh(1)$ does not determine $\Oh(i)$. Orbifold curves are pretty simple
(see Section~\ref{sec!Cur}); however, to handle orbifold behaviour in
codimension~1 the graded structure sheaf $\bigoplus\Oh(i)$ must be
specified. There are in any case theoretical advantages in thinking of
$\bigoplus\Oh(i)$ systematically as a graded structure sheaf.
The final condition is stated here for $n\ge2$, when $\om_X$ is a
divisorial sheaf. More care is needed to handle orbifold behavior in
codimension~1: namely, rather than a single dualising sheaf $\om_X$, we
need the graded dualising sheaf of the graded structure sheaf
$\bigoplus\Oh_X(i)$. The case $n=1$ (orbifold curves) can be treated in
terms of fractional divisors $A=\sum\frac{a_i}{r_i}P_i$ and the orbifold
canonical class $K_{C,\orb}=K_C+\sum\frac{r_i-1}{r_i}P_i$; see
\ref{ssec!curve}, \cite{De} and \cite{W}.
\end{rmk}}
\begin{rec} \label{rec!Qdiv} \rm Replace $A$ by a $\Q$-divisor that
includes the fractional term $\frac brP$ for each orbifold point $P$ of
type $\frac1r(a)$ on $C$, where $b$ is the inverse of $a$ modulo $r$;
that is, write
\[
A=A_0+\sum_\sB \frac brP \quad \hbox{with $A_0$ an
integral divisor.}
\]
At the same time, replace $K_C$ by $K_{C,\orb}=K_C+\sum_\sB\frac{r-1}rP$.
Then $\Oh_X(i)=\Oh_X(iA):=\Oh_X([iA])$, Serre duality takes the correct
form, and the graded ring $R(C,A)$ is Gorenstein with canonical weight
$k_C\in\Z$ if and only if $K_{C,\orb}=k_CA$. \end{rec}
\paragraph{Discussion} We explain why the construction is right, leaving
the details to you (see Demazure \cite{De} and Watanabe \cite{W}); this
also appears subliminally in many places in papers by Kawamata, Reid,
Shokurov, and others.
The assumption on $C$ is that at each orbifold point, the local
parameter $z_P$ of the complex curve $C$ is $z_P=w_P^r$, where $w_P$ is
the {\em orbinate} (orbifold coordinate, that is, the coordinate on the
overlying orbifold cover), and the $\mu_r$ action is $w_P\mapsto \ep^a
w_P$. The sheaf $\Oh_C(i)$ consists locally of the $i$th eigensheaf of
this action; that is, it consists of monomials $w_P^m$ with $m\ge0$ and
$am\equiv i$ mod $r$, that is $m\equiv bi$ mod $r$, together with power
series consisting of sums of these (formal, convergent or algebraic
power series, according to taste). Therefore the sheaf $\Oh_C(i)$ is
based by $w_P$ to the $[\frac{bi}r]$ power. In the same way, the
orbifold canonical class is based by $\dd w_P$, which is equal to a
constant times $w_P^{\frac{r-1}r}\dd z_P$.
\begin{exc} \rm If $C$ is an orbifold curve with a basket
$\sB=\{P,\rth(a)\}$, and $\bigoplus\Oh_C(i)$ is represented by $\Oh(iA)$
as just discussed, then $R(C,\bigoplus\Oh_C(i))$ is Gorenstein with
index $k_C$ if and only if $a+k_C\equiv 0$ mod $r$ for each $P\in\sB$,
and then its Hilbert series is given by
\begin{equation}
P_C(t)=P_I+\sum_{\sB}P_\orb(\rth(a),k_C)
\end{equation}
where as in Theorem~\ref{thm!first}, the initial term $P_I$ is
$\frac1{(1-t)^2}$ times a Gorenstein symmetric polynomial of
degree~$k_C+2$. Each orbifold term is obtained by calculating $b$, the
inverse of $a$ mod $r$, and taking $\frac{1-t^b}{1-t}$ modulo
$\frac{1-t^r}{1-t}$ written out as a Laurent polynomial supported in the
appropriate interval. \end{exc}
\begin{exa} \rm Let $C=\PP^1$ and take $A=\frac35P+\frac12Q-R$ (where
$P,Q,R\in C$ are distinct points); then
$K_{C,\orb}=\frac45P+\frac12Q-2R$, so $-7A\sim K_{C,\orb}$. One sees
that $R(C,A)=k[x,y]$ where $\wt x,y=2,5$. In degree~2, $x$ vanishes at
$P$ to order $\frac15$, so $P=(0,1)$ is a $\frac15(2)$ orbifold point
with $x$ as orbinate. Note that $\frac1{10}=\frac35+\frac12-1$ and
\[
P(C,A)=\frac1{(1-t^2)(1-t^5)}
=\frac{t^{-2}}{(1-t)(1-t^2)}+\frac{-t^{-2}-t}{(1-t)(1-t^5)}
\]
The case of $\PP^1(a,b)$ with any coprime $a,b$ is similar. \end{exa}
\begin{exa} \rm The weighted projective line $X=\PP(2,5)$ has orbifold
points of type $\frac15(2)$ and $\frac12(5)=\frac12(1)$, in the global
context $k_X=-7$. The initial term $P_I=0$ whenever $c<0$. The orbifold
terms are \begin{multline}
P_{\per}({\textstyle\frac15(2)})=\frac1{1-t^5}\times
\frac15\times(-3t-t^2-4t^3-2t^4), \\ P_{\tot}({\textstyle\frac15(2)},-7)
=P_{\per}+\frac{-t^{-2}+t\1-(2/5)t}{(1-t)^2}
=\frac{-t^{-2}-t}{(1-t)(1-t^5)}\,, \label{eq!35th} \end{multline} and
\begin{multline} P_{\per}({\textstyle\frac12(1)})=\frac1{1-t^2}\times
\frac12\times(-t), \\ P_{\tot}({\textstyle\frac12(1)},-7)
=P_{\per}+\frac{t^{-2}-t\1+1-(1/2)t}{(1-t)^2}
=\frac{t^{-2}}{(1-t)(1-t^2)}\,. \label{eq!12th} \end{multline} Adding
these gives
\begin{equation}
\frac{-t^{-2}-t}{(1-t)(1-t^5)}+\frac{t^{-2}}{(1-t)(1-t^2)} =
\frac1{(1-t^2)(1-t^5)}\,.
\end{equation}
Although mystifying at first sight, these calculations are really very
easy, and understanding them illuminates the general case. In
(\ref{eq!35th}), the terms
$\frac1{1-t^5}(-\frac35t-\frac15t^2-\frac45t^3-\frac25t^4)$ are the
periodically repeating fractional parts lost on rounding down
$\Oh_C(\frac{3i}5P)$. Since the plurigenera are integers, We must add
some global term to compensate (thus adding to the degree in the
polynomial part of RR); adding $\frac35t$ in the numerator of
$P_{\grow}$ would give \begin{multline}
\frac1{1-t^5}\left(-\frac35t-\frac15t^2-\frac45t^3-\frac25t^4\right)
+\frac35\times \frac{t}{(1-t)^2} \\
=t^2+t^3+2t^4+3t^5+3t^6+4t^7+4t^8+5t^9+6t^{10}+ \cdots\\ =
\frac{t^2+t^4+t^5}{(1-t)(1-t^5)}\,, \end{multline} the natural integral
growth of $[\frac{3i}5]$, incrementing when $i\equiv2,4$ or $0$ mod~5.
However, in the context $k=-7$, we want the contribution to be symmetric
of degree $-7$ in the sense of the functional equation (\ref{eq!fn}). So
rather than add $\frac35t$ in the numerator we add
$-t^{-2}+t\1-\frac25t$; the numerator $-t^{-2}-t$ in (\ref{eq!35th}) is
symmetric of degree $-1$, so that the whole contribution $P_{\tot}$ has
degree $-7$.
In (\ref{eq!12th}) the numerator $t^{-2}$ is symmetric of degree $-4$
(because the only term $t^{-2}$ is the centre of the symmetry), so
$P_{\tot}$ again has degree $-7$. \end{exa}
\subsection{Isolated orbifold singularities} Now let $X$ be projectively
Gorenstein with isolated orbifold singularities, with $k_X$ and $c$ as
in Definitions~\ref{defn!pGor}--\ref{defn!coind}. The ingredients in the
plurigenus formula are as follows: \begin{itemize} \item the dimension
$n$; \item the canonical weight $k_X$ (see Definition~\ref{defn!pGor});
\item the coindex $c=k_X+n+1$; \item the first $[c/2]$ plurigenera
$P_i(X)$ for $i=1,\dots,[c/2]$; \item a basket $\sB$ of isolated
orbifold points $\sB=\{\rth(a_1,\dots,a_n)\}$, with $r\ge2$. \item[] For
an orbifold point $\rth(a_1,\dots,a_n)$, the isolated assumption is that
each $a_1,\dots,a_n\in[1,r-1]$ is coprime to $r$. \item[] Our assumption
that $X$ is projectively Gorenstein with $\om_X=\Oh_X(k_X)$ implies that
each $\rth(a_1,\dots,a_n)\in\sB$ satisfies
\begin{equation}
k_X+\sum_{j=1}^n a_j \equiv 0 \quad\hbox{mod $r$.}
\end{equation}
\end{itemize}
We now use these ingredients to cook up an {\em initial term}
$P_{I,X}(t)$, and for each point $\rth(a_1,\dots,a_n)\in\sB$ an {\em
orbifold contribution} $P_\orb(\rth(a_1,\dots,a_n),k_X)$, each computed
by a simple recipe, so that
\begin{equation}
P_X(t)=P_{I,X}(t)+\sum_{\sB}P_\orb\bigl(\rth(a_1,\dots,a_n),k_X\bigr).
\end{equation}
Note that ``initial term'' certainly does not mean ``leading term'': it
fixes up the initial plurigenera, but not the leading order of growth.
\begin{defn}[Initial term] \rm The {\em initial term} $P_{I,X}(t)$ is
\begin{equation}
P_{I,X}(t)=\frac{A(t)}{(1-t)^{n+1}}\,,
\end{equation}
where $A(t)$ is a symmetric polynomial of degree $c$ with integer
coefficients, uniquely determined by the condition that the formal power
series $P_{I,X}$ has the given $P_i(X)$ as coefficient of $t^i$ up to
$i=[c/2]$ (with $P_0=1$ if $c\ge0$).
If $c<0$ then also $[c/2]<0$, and $P_I=0$.
\end{defn}
\begin{rec} \rm \begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})} \item Set
$A_0=\sum_{i=0}^{[c/2]} P_it^i$ (this is 0 is $c<0$);
\item set $A_1=(1-t)^{n+1}A_0$ and $P'_i=$ coefficient of $t^i$ in $A_1$
for $i=0,\dots,[c/2]$;
\item finally, set $A(t)=\sum_{i=0}^c P''_it^i$ where $P''_i=P'_i$ or
$P'_{c-i}$. \end{enumerate} \end{rec}
\begin{exa} \rm Take $n=3$, $c=5$, $P_1=3$, $P_2=7$, $A_0=1+3t+7t^2$;
then $A_1=(1-t)^4A_0=1-t+t^2+\cdots$, so that
$P_I(t):\frac{1-t+t^2+t^3-t^4+t^5}{(1-t)^4}$; you check that
$P_I(t)=\frac{1+2t^3+t^6}{(1-t)^3(1-t^2)}$ is the Hilbert series of the
nonsingular canonical 3-fold $X(6,6)\subset\PP(1,1,1,2,3,3)$. \end{exa}
\begin{defn}[Orbifold term] \rm Let $\rth(a_1,\dots,a_n)$ and $k_X$ be
as above. Its orbifold contribution is defined by
\begin{equation}
P_\orb\bigl(\rth(a_1,\dots,a_n),k_X\bigr) = \frac{B(t)}{(1-t)^n(1-t^r)},
\end{equation}
where the numerator $B(t)$ is
\begin{itemize}
\item the inverse modulo \ $\displaystyle \frac{1-t^r}{1-t} =
1+t+\cdots+t^{r-1}$ \ of \ $\displaystyle
\prod\left(\frac{1-t^{a_i}}{1-t}\right)$
\item as a Laurent polynomial with support in $[\ga+1,\ga+r-1]$, where
$\ga=[c/2]$.
\end{itemize}
\end{defn}
\begin{fn}\label{fn!Qorb}
\begin{verbatim}
function Qorb(r,LL,k)
L := [ Integers() | i : i in LL ]; // allows empty list
if (k + &+L) mod r ne 0 then
error "Error: Canonical weight not compatible";
end if;
n := #LL;
Pi := &*[ R | 1-t^i : i in LL];
h := Degree(GCD(1-t^r, Pi));
// degree of GCD(A,B) // -- simpler calc?
l := Floor((k+n+1)/2+h);
// If l < 0 we need a kludge to avoid programming
// genuine Laurent polynomials
de := Maximum(0,Ceiling(-l/r));
m := l + de*r;
A := (1-t^r) div (1-t);
B := Pi div (1-t)^n;
H,al_throwaway,be:=XGCD(A,t^m*B);
return t^m*be/(H*(1-t)^n*(1-t^r)*t^(de*r));
end function; \end{verbatim}
\end{fn}
Given $\frac1r(a_1,\dots,a_n)$, in the context of $K_X = kA$, the
calculation is the Euclidean algorithm for the hcf of
\begin{equation}
A := \frac{t^r-1}{t-1}=1+t+\cdots+t^{r-1}, \quad\hbox{and}\quad
B := \prod \frac{t^{a_i}-1}{t-1}.
\end{equation}
Write $h=\hcf(A,B)$ (which is 1 in the current case, since the $a_i$ are
coprime to $r$) and $l=\bigl[\frac{k+n+1}{2}\bigr]+\deg h$. (The $l$
just translates the support of the Laurent polynomial.) Now calculate
the hcf by the Euclidean algorithm in the form
\begin{equation}
\hcf(A,t^lB) =: H = \al A+\be t^lB,
\end{equation}
and return $\frac{\be t^l}{H(1-t)^n(1-t^r)}$. Since $\be$ is in the
range $[0,r-2]$ it follows that $\be+l$ is in the range $\bigl[ l,l+r-2
\bigr]$, as required.
\subsection{Some progress on Hilbert series of CY orbifolds} The general
formula is
\[
P_I + \sum_{B\in\sB} P_\orb(B,0) + \sum_{C\in\sC} P_C(C)
\]
here (1) the {\em initial term} is
\[
P_I=\frac{1+at+bt^2+at^3+t^4}{(1-t)^4}.
\]
In the point sum, each $B=\rth(a_1,a_2,a_3)$, and the term is
\[
P_\orb(B,0) = \frac{\hbox{Num}}{(1-t)^3(1-t^r)}
\]
where, in the isolated case, Num is the unique polynomial with support
in $[3,r]$ which is the inverse of $\prod_{i=1}^n \frac{1-t^ai}{1-t}$;
in general you also have to take out $1/(1-t^b)$ for common factors,
etc., and the function is the Magma function Qorb,
Function~\ref{fn!Qorb}.
In the curve sum, each $C$ is of the form $1/r(a,r-a)$ plus extra data,
and the term is
\[
P_C=\frac{\hbox{Num}}{(1-t)^2(1-t^r)^2}=\frac{\hbox{Num}}{[1,1,r,r]}
\]
where Num is a symmetric polynomial of degree $2r+2$ supported in
$[3,2r-1]$. Num has $r-1$ arbitrary coefficients, and the initial
expectation is that all values in an open range will occur.
\begin{rmk}[Preliminary notes to myself] \label{rmk!Ga} \rm I think I
have progress on the 1-dim orbifold locus contributions, Namely, if $X$
has transverse $\frac1r(L)$ singularities along a curve $\Ga$, in the
context of $K_X=\Oh_X(k)$, the contribution is
\begin{equation}
A\times P_\orb(\frac1r(L),k+r)\times\frac1{(1-t^r)} +B\times
\frac{t^a}{(1-t)^n(1-t^r)} \label{eq!star}
\end{equation}
where $A$ and $B$ are Gorenstein symmetric Laurent polynomials of given
degree and support. I'm not quite sure, but one prediction is that
``most'' $A$ and $B$ within some range occur; but maybe there are
divisibility or congruence conditions, or at the other extreme, only one
or two $A$ and $B$ allowed. Each of $A$ and $B$ has approx $(r-1)/2$
free coefficients, which is reasonable since they contain implicitly the
RR data for $\Ga,\Oh_\Ga(i)$, the normal bundle to $\Ga$ (with its
$\Z/r$ eigendecomposition) and all their twists. In any case, any choice
of $A$ and $B$ give rise to Hilbert series with the right symmetry, so
we can make 10 billion baskets of CYs in the very near future.
Let's try to say that more precisely. $P_\orb(\frac1r(L),k)$ is the
contribution of a single point $\frac1r(L)$ on an $m$-fold, not
necessarily isolated, where $m=\#L$. It has denominator $(1-t)^m(1-t^r)$
and is Gorenstein symmetric of degree $k$, and its numerator $\al$ is a
Laurent polynomial of support of length $ AA[34];
[ 1, 13, 28, 70, 111, 223 ]
> Bask(AA[34]);
[
[ 13, 1, 5, 7 ],
[ 28, 1, 13, 14 ],
[ 70, 1, 28, 41 ],
[ 111, 13, 28, 70 ]
]
[
[ 14, 1, 13 ]
]
> X(AA[34]);
(-t^22-t^20-t^18-t^17-t^16-t^15-t^14-t^13-t^12-t^10-t^8)
/(t^30-2*t^29+t^28-2*t^16+4*t^15-2*t^14+t^2-2*t+1)
> PartialFractionDecomposition(X(AA[34])/t^3*(1-t)^4);
[
,
,
,
,
]
> [X(A) eq &+[K| x : x in PC(A)] : A in AA];
// gives "true" 42 times, checking the routines don't crash.
\end{verbatim}
The above analyses existing examples, and can be quickly programmed to
analyse the Hilbert series of the 7555 hypersurfaces. I want to use
these ideas to predict new examples. The nicely generated CYs are sparse
in the set of all possible Hilbert series, so that if I modify $P_Y(t)$
by a clumsy amount, it is unlikely that the new $P_Y$ will correspond to
a $Y$ that we can work with. However, I can modify the above
\verb! Num_r ! to make more delicate variations, e.g.,
\begin{verbatim}
> A := [1,4,5,5,5,20];
\end{verbatim}
is a hypersurface with curve $\Ga$ of $1/5(1,4)$ contributing
\[
(2t^9 + 4t^8 + 6t^7 + 2t^6 + 6t^5 + 4t^4 + 2t^3)/[1,1,5,5]
\]
I modify it by changing the $6,2,6$ in the middle to $5,4,5$ (which should
remove a generator in deg~5, and leave the point singularities and the
degree of the curve unchanged).
\begin{verbatim}
> P1 := (1-t^20)/Denom([1,4,5,5,5]);
// the Hilbert series of Y(20) in PP(1,4,5,5,5)
> P2 := P1 + (-t^7+2*t^6-t^5)/Denom([1,1,5,5]);
> P2*Denom([1,4,5,5,6,9]);
t^30 - t^18 - t^12 + 1
\end{verbatim}
That is, the modification gives $Y(12,18)\subset\PP(1,4,5,5,6,9)$.
Or change the $4,6,2,6,4$ in the middle to $3,6,4,6,3$ (which should
remove a generator in deg~4 and add one in deg~6).
\begin{verbatim}
> P3 := P1 + (-t^8+2*t^6-t^4)/Denom([1,1,5,5]);
\end{verbatim}
gives a plausible codim~4 guy with $9\times16$ resolution.
Or change the Numerator of the $\Ga$ term to
\[
(2t^9 + 3t^8 + 7t^7 + 2t^6 + 7t^5 + 3t^4 + 2t^3)
\]
\begin{verbatim}
> P4 := P1 - (t^8-t^7-t^5+t^4)/Denom([1,1,5,5]);
> P4*Denom([1,5,5,5,7,8,9]);
-t^40 + t^26 + t^25 + t^24 + t^23 + t^22 - t^18 - t^17 - t^16
- t^15 - t^14 + 1
\end{verbatim}
gives codim~3 candidate $Y\subset\PP(1,5,5,5,7,8,9)$ with $5\times5$
Pfaffian matrix of degrees
\begin{verbatim}
5,6,7,8
7,8,9
9,10
11
\end{verbatim}
We can play this game about a million times with the existing list.
\begin{verbatim}
> load "KS2"; // Loading "KS2"
> #AAKS2; // 14817
function BB(AA,r,a)
// find transverse 1/r(a,r-a) curves in list AA of CYs
BB0 := [A : A in AA
| (([r,a,r-a] in C) or ([r,r-a,a] in C)) where B,C is Bask(A)];
BB1 := [BB0[i] : i in [1..#BB0]
| &and[GCD(b[1],r) eq 1 : b in Bask(BB0[i])]];
return [BB1[i] : i in [1..#BB1] | (BB1[i][6] mod r) in [0,a,r-a]];
end function;
> AA := BB(AAKS2,5,2);
> #AA; // there are 19 of them
\end{verbatim}
\section{More ice cream}\label{sec!Cur}
The second term in \cite{BSz}, Cor 3.3 is the sum
\begin{verbatim}
function Second(r,k)
return &+[ ikbar*(r-ikbar)*(r-2*ikbar)*t^i
where ikbar is (i*k mod r) : i in [1..r-1]] / (6*r*(1-t^r));
end function;
\end{verbatim}
I subtract a little initial term from it to get rid of term in $t$ and
$t^2$, of the form
\[
\frac{at + bt^2 + at^3}{(1-t)^4}.
\]
\begin{verbatim}
Second(r,k)-(r-2*k)*(r-k)*k*(t+t^3)/6/r/(1-t)^4
+((r-2*k)*2*ka*(r+2*ka)*t^2/6/r/(1-t)^4
where ka is Min(k,r-k));
\end{verbatim}
This gives nice quantities like
\begin{align*} 5,2 & \mapsto \frac{3t^5 - 2t^4 + 3t^3}{[1,1,1,5]}
\quad\hbox{and} \\
8,3 & \mapsto \frac{(8t^7-12t^6+17t^5-12t^4+8t^3)(1+t)}{[1,1,1,8]}
\end{align*}
Abstracting from that gives function called Nterm, of the form
\[
\frac{\hbox{integral symm polynom supported in
[3..r]}}{[1,1,1,r]}
\]
with numerator Num uniquely determined by the condition
\[
\hbox{Num}*(1-t^b)^2(1-t^{r-b})/(1-t)^3 \equiv 1 + t^b\mod (1-t^r)/(1-t)
\]
(note the side-step $k\mapsto b=$ inverse of $k$ mod $r$). It would be
jolly convenient to be able to calculate directly in the ring
$\Q[t]/((1-t^r)/(1-t))$. That is,
\begin{verbatim}
function Nterm(r,b)
return t^2*(t^(r-2)*(1+t^b)*InverseMod(Denom([b,b,r-b]) div (1-t)^3,
((1-t^r) div (1-t))) mod ((1-t^r) div (1-t)))/Denom([1,1,1,r]);
end function;
\end{verbatim}
The $t^2*t^{r-2}$ is just a device for shifting the support into the
interval $[3,\dots,r]$. I can test this as much as I like:
\begin{verbatim}
for i in [1..20] do r := Random(500); k := Random(r);
if GCD(r,k) eq 1 then b := InverseMod(k,r);
r,k,Second(r,k)-(r-2*k)*(r-k)*k*(t+t^3)/6/r/(1-t)^4
+ ((r-2*k)*2*ka*(r+2*ka)*t^2/6/r/(1-t)^4
where ka is Min(k,r-k)) eq Nterm(r,b);
end if;
end for;
\end{verbatim}
(That looks trivially easy, but it took me 3 days of arm-wrestling with
the computer to get it to work.) I think the $\deg\Ga$ terms are
basically simpler; I hope this concludes the treatment of pure
$1/r(a,r-a)$ curves.
\section{More notes on ice-cream}
The two methods we have are working with are guessing from families of
examples, and figuring out how to sum the expressions in \cite{B},
\cite{BSz} in closed form. For the former consider
$\PP(r,\dots,r,a_1,a_2,\dots,a_n)$ with $r$ repeated $d+1$ times and the
$a_i$ coprime to $r$ and to each other. This has a pure locus $\PP^d$ of
transvers type $1/r(a_1,a_2,\dots,a_n)$ and only isolated points Then
you can subtract off all the junk, just leaving you with the
contribution of the pure $1/r$ locus (of any dimension, any transverse
cyclic orbifold). This lets us step back a bit from the $N_C$ term of
\cite{B}, \cite{BSz}, which is possibly confusing because it treats the
two isotypical normal bundles as sum and difference, rather than two
separate terms. Experiments suggest that the contributions should be
some kind of compound ice cream function (maybe double cone), with the
main ingredients, as with $P_\orb$, derived from things like
\verb! InverseMod(ph(i), ph(r)) !.
For example $\PP(r,r,j)$ with $r$ coprime to 3 and $j\equiv3$ mod $r$.
Set $X$ to be the actual Hilb series minus the $P_\orb$ term for the
isolated point $1/j(r,r)$. I treat $X$ as the unknown to be investigated.
Subtract off the term \verb! Qorb(r,[j],kP+r)/(1-t^r) ! that corresponds
to cutting by a zero-dim section of the $\PP^1$, and is the only part of
the formula with denominator $\prod_{a\in[1,r,r]}(1-t^a)$.
\begin{verbatim}
X := 1/Denom(A) - Qorb(j,[r,r],kP) - Qorb(r,[j],kP+r)/(1-t^r);
\end{verbatim}
\begin{exa} \rm The following function calculates the contribution of a
pure curve $\PP(r,r)$ of transverse type $1/r(3)$ for $r=1$ or 5 mod 6.
(The given routine is for $\PP(25,25,103)$.)
\begin{verbatim}
// i = 3 mod r, works for r = 1 or 5 mod 6
r:=25; a:=2; i:=3; j:=2*r*a+i; A:=[r,r,j]; kP := -&+A;
X:=1/Denom(A)-Qorb(j,[r,r],kP);
X-Qorb(r,[j],kP+r)/(1-t^r)
+ ( -(-1)^((r mod 6) div 3) *
InverseMod(t^((r+1) div 2)*ph(3),ph(r))
+ a*((1+t^3)*InverseMod(t*ph(3)^2,ph(r)) mod ph(r)) )
/t^(r*(a+1)-1)/Denom([1,1,r]);
\end{verbatim}
Practically the same routine works for $r=2$ or 4 mod 6, with the term
\verb!InverseMod(t^((r+1) div 2)*ph(3),ph(r)) !
replaced by
\[
\verb! ((1+t)*InverseMod(t^((r+2) div 2)*ph(3),ph(r)) mod ph(r))!.
\]
There should be a more systematic solution without the case division.
\end{exa}
\subsection{Preliminary draft:} You get functions with numerators that for
\verb! PP(11,11,376) ! look like:
\[
154t^9+17t^8+120t^7+51t^6+86t^5+86t^4+51t^3+120t^2+17t+154
\]
This a sum of two arithmetic progressions with step $t^2$ starting out
from the ends. Dividing into classes mod $2r$ gives them as sums of two
more-or-less sensible terms: write $r=r_1+r_2$ with $r_i=\frac{r\pm1}2$.
Then the first term is $(1\pm t^{r_1})(1\mp t^{r_2})/(1-t^2)$. The
second is something that sums in closed form like
\[
\frac{-9t^{12}-10t^{11}+t^{10}-t^2+10t+9}{(1-t^2)(1+t)}.
\]
I hope we'll eventually get this in into a more convincing form.
\begin{verbatim}
r:=11; a:=17; i:=2; j:=2*r*a+i;
A:=[r,r,j]; kP := -&+A; X:=1/Denom(A)-Qorb(j,[r,r],kP);
X - Qorb(r,[j],kP+r)/(1-t^r)
+ ((&*[1-(-t)^a : a in L] div (1-t^2) where L:=[(r+1) div 2,(r-1) div 2])
+ a*(((r-2)*(1-t^(r+1))+(r-1)*(t-t^r)-(t^2-t^(r-1))) div
+((1-t^2)*(1+t))))/t^((a+1)*r-1)/Denom([1,1,r]);
\end{verbatim}
That did $\PP(11,11,375)$, but it works for all odd $r$ and all $a$.
\begin{thebibliography}{GW}
\bibitem[B]{B} A. Buckley: Orbifold Riemann-Roch for 3-folds and
applications to Calabi-Yaus, Univ. of Warwick PhD thesis, viii + 119
pp., get from \verb! http://www.fmf.uni-lj.si/~buckley !
\bibitem[BSz]{BSz} A. Buckley and B. Szendr{\H o}i: Orbifold
Riemann-Roch for threefolds with an application to Calabi--Yau geometry,
to appear in J. Algebraic Geom., preprint math.AG/0309033, 19~pp.
\bibitem[De]{De} Michel Demazure, Anneaux gradu\'es normaux, in
Introduction \`a la th\'eorie des singularit\'es, II, Hermann, Paris
(1988) pp.~35--68
\bibitem[GW]{GW} S. Goto and K-i Watanabe, On graded rings. I, J. Math.
Soc. Japan {\bf30} (1978) 179--213
\bibitem[Mu]{Mu} MUKAI Shigeru, Biregular classification of Fano 3-folds
and Fano manifolds of coindex~3, Proc. Nat. Acad. Sci. U.S.A. {\bf86}
(1989) 3000--3002
\bibitem[YPG]{YPG} M. Reid, Young Person's Guide to canonical
singularities
\bibitem[W]{W} WATANABE Kei-ichi, Some remarks concerning Demazure's
construction of normal graded rings, Nagoya Math. J. {\bf 83} (1981)
203--211
\end{thebibliography}
\end{document}
The statement for a pure m-dim orbifold locus of transverse type 1/r[L]
in the context of K = kA is
P = P_Initial + P_Orb
where P_Orb is a sum of
P_i / (1-t)^{i+1} * (1-t^r)^{m-i}
and the P_i are Laurent polynomials, the same as in the 0-dim orbifold
formula shifted to a suitable interval to give right symmetric degree.
Surface S13 in PP(1,1,4,4) has curve Ga of 1/4(1) and k = 3 so c = 6.
> PI := (1-t-t^5+t^6)/(1-t)^3;
> A := (t^4+t^5)/&*[1-t^i : i in
[1,1,4]];
> B := (-t^5-t^6-t^7)/&*[1-t^i : i in [1,4,4]];
>
(PI+2*A+B)*&*[1-t^i : i in [1,1,4,4]]; -t^13 + 1
You can try out PI+a*A+b*B in a range of values of a,b; e.g.,
> (PI+A+B)*&*[1-t^i : i in [1,1,4,8,12]]; t^29 - t^24 - t^5 + 1 Predicts
S(5,24) in PP(1,1,4,8,12). Note that genus Ga = 1
> (PI+3*A+B)*&*[1-t^i : i in [1,1,4,4,4,5]]; -t^22 + 2*t^14 + 2*t^13 +
t^12 - t^10 - 2*t^9 - 2*t^8 + 1 A Pfaffian 3 3 4 4 \\ 4 5 5 \\ 5 5 \\ 6.
I think that Ga is PP(z1,z2) and PP(z3) is an isolated nonquasismooth
point. Maybe a mirage.
> (PI+3*A+2*B)*&*[1-t^i : i in [1,1,4,4,4]]; t^17 - t^9 - t^8 + 1
Predicts S(8,9) in PP(1,1,4,4,4)
> (PI+3*A+3*B)*&*[1-t^i : i in [1,1,4,4,4]]; t^17 - t^12 - t^5 + 1
Predicts S(5,12) in PP(1,1,4,4,4)
Here b = deg Ga the orbifold curve 1/4(1), whereas increasing a (?)
possibly increments genus(Ga) or adds isolated or embedded points
In the same way, B is easy to understand as Porb(4,[1],6)/(1-t^4),
whereas A with its factor of (1+t) is more subtle.
There is also a hint that B only occurs in the comb A+B =
(t^4+t^5+t^6+t^7+t^8)/ denom[1,4,4].
-------------------------
// Test 1: when n = 1, this is just a translate of Scores(b/r).
[Porb(13,[i],13-i) : i in [1..12]];
/* gives t^13/(t^14 - t^13 - t + 1), .. (t^13 + t^12 + t^11 + t^10 +
t^9 + t^8 + t^7 + t^6 + t^5 + t^4 + t^3 + t^2)/(t^14 - t^13 - t + 1) */
// Test 2: PP(7,9,23) has 1/7(2,2), 1/9(7,5), 1/23(7,9) in k = -39 //
and PP(3,4,5,7) has k = -19
Porb(7,[2,2],-39) + Porb(9,[5,7],-39) + Porb(23,[7,9],-39); $1 eq
1/&*[1-t^i : i in [7,9,23]];
Porb(3,[1,2,1],-19) + Porb(4,[3,1,3],-19) + Porb(5,[3,4,2],-19) +
Porb(7,[3,4,5],-19); $1 eq 1/&*[1-t^i : i in [3,4,5,7]];
// Test 3: Fano with k = -2 PI := (1-2*t+t^2)/(1-t)^4; X := PI +
Porb(5,[1,2,4],-2); X*&*[1-t^i: i in [1,1,2,3,4,5]]; // t^14 - t^8 - t^6
+ 1
compare the formula [BSz] Cor 3.3 with my Qorb
Their formula is a double sum, the X below. It is normalised to
-Qorb(r,[a,b,c],0) by adding an initial term.
r,a,b,c with a+b+c == 0 mod r
Qorb(r,[a,b,c],0)
r:=41; a:=1; b:=1; c:=r-a-b; KK := CyclotomicField(r); RR` :=
PolynomialRing(KK); X:=1/r*1/(1-t^r)*R!(&+[&+[
(ep^(-i*j)-1)/(1-ep^(-a*j))/(1-ep^(-b*j))/(1-ep^(-c*j)) : j in [1..r-1]
| (a*j mod r ne 0) and (b*j mod r ne 0) and (c*j mod r ne 0) ]*tt^i : i
in [1..r-1]]); r*(X+Qorb(r,[a,b,c],0));
S!(r*(X+Qorb(r,[a,b,c],0)));
Assertion: the \wP_Q(t) of [BSz], Cor 3.3 is
-Qorb(r,[a,b,c],0) - 1/r*(al*t+be*t^2+al*t^3)/(1-t)^4; where al, be is
of the following kind:
1,1,11 -> al, be = 7, -20
be+4*al eq 8 be + 3*al eq 1 1,2,10 ->
al, be = 3, -8
be+4*al eq 4 be + 3*al eq 1 1,3,9 -> al, be = 2,
-5
be+4*al eq 3 be + 3*al eq 1 1,4,8 -> al, be = -1, 4
be+4*al eq 0 be + 3*al eq 1 1,5,7 -> al, be = 4, -11
be+4*al eq
1 be + 3*al eq 1 1,6,6 -> al, be = -8, 25
be + 3*al eq 1 2,2,9
-> al, be = -1, 11
be + 3*al eq 8 2,3,8 -> al, be = 1, -5
be + 3*al eq -8 2,4,7 -> al, be = 0, 3
be + 3*al eq 3
2,5,6 -> al, be = -3, 14
be + 3*al eq 5 3,3,7 -> al, be = -7,
34 3,4,6 -> al, be = -5, 21 3,5,5 -> al, be = -6, 34 4,4,5 -> al, be =
-10, 39
first unknown al, be is (r^2-1)/24, (r-1)(r+7)/12
1/7(1,1,5): 2,-7 1/7(1,2,4): 1,-3 1/7(1,3,3): -1,5
1/11(1,1,9): 5,-15 1/11(1,2,8): 1,0 1/11(1,3,7): 0,1
1/13(1,1,11): 7,-20 1/13(1,2,10): 3,-8 1/13(1,3,9): 2,-5
1/17(1,1,15): 12,-32 1/17(1,2,14): 3,-3 1/17(1,3,13): 5,-17
1/19(1,1,17): 15,-39 1/19(1,2,16): 6,-15 1/19(1,3,15): -1,10
1/23(1,1,21): 22,-55 1/23(1,2,20): 6,-8 1/23(1,3,19): 2,-2
1/29(1,1,27): 35,-84 1/29(1,2,26): 10,-15 1/29(1,3,25): 11,-34
1/31(1,1,29): 40,-95 1/31(1,2,28): 15,-35 1/31(1,3,27): 1,11
1/37(1,1,35): 57,-132 1/37(1,2,34): 21,-48 1/37(1,3,33): 12,-27
1/41(1,1,39): 70,-160 1/41(1,2,38): 21,-35 1/41(1,3,37): 19,-55
1/43(1,1,41): 77,-175 1/43(1,2,40): 28,-63 1/43(1,3,39): 5,8
1/47(1,1,45): 92, -207 1/47(1,2,44): 28, -48 1/47(1,3,43): 12,-20
1/53(1,1,51): 117,-260 1/53(1,2,50): 36,-63 1/53(1,3,49): 29,-80
1/57(1,1,55): 406/3,-896/3
1/59(1,1,57): 145,-319
`