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%% Orbifold Riemann--Roch and plurigenera
%% Miles Reid
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\begin{document}
\title{Orbifold Riemann--Roch and plurigenera}
\author{Miles Reid}
\date{}
\maketitle
\begin{abstract}
I give a general formula for the Hilbert series of a polarised
$n$-dimensional orbifold (for example, 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 my work over 20 years to a few lines of computer algebra.
\end{abstract}
For dramatic effect, I state a simple case of the theorem first,
leaving definitions and explanations for later.
\begin{theorem} \label{thm!first}
Let $X,\bigoplus\Oh_X(i)$ be a simply polarised $n$-fold with
$n\ge2$. Assume that $X$ is projectively Gorenstein with canonical
weight $k_X$ and has a basket of isolated orbifold points
$\sB=\{\rth(a_1,\dots,a_n)\}$ 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 The initial term $P_I(t) = \frac{A(t)}{(1-t)^{n+1}}$ has
denominator $(1-t)^{n+1}$, and its numerator $A(t)$ is a
Gorenstein symmetric polynomial of degree $c=k_X+n+1$ (the {\em
coindex} of $X$). It determines and is determined by the
$\left[\frac c2\right]$ initial terms of $P_X(t)$ (and is $0$ if
$c<0$).
\item Each orbifold term $P_\orb(\rth(a_1,\dots,a_n),k_X)
=\frac{B(t)}{(1-t)^n(1-t^r)}$ has denominator $(1-t)^n(1-t^r)$,
and numerator $B(t)$ the unique 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}
\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{itemize}
\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{itemize}
I say {\em simply} polarised to mean $\Z$-graded or $\N$-graded (as
opposed to lattice polarised or graded by a more complicated semigroup).
\end{defn}
I 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, and \ref{ssec!codim01} in general.) 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 my 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, I 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 $P_i(X)$. My problem is to
calculate $P_X(t)$ under extra conditions.
\end{defn}
\begin{defn}\label{defn!pGor} \rm I 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 $X$.
My 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{The curve case} \label{ssec!curve}
The initial statement of Theorem~\ref{thm!first} does not cover
the elementary case of a curve $C$ with isolated orbifold points.
The simple device used throughout \ref{ssec!term} of avoiding the
full graded structure sheaf of $X,\Oh_X(i)$ by writing
$\Oh_X(i)=\Oh_X(iA)$ with $A$ a $\Q$-Cartier Weil divisor 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 is simple and fairly well known.
\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
$K_{C,\orb}=K_C+\sum_\sB\frac{r-1}r$.
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 if and
only if $K_C=k_CA$ for some integer $k_C$.
\end{rec}
\paragraph{Discussion} I explain why the construction is right,
leaving the details to you (see Demazure \cite{De} and Watanabe
\cite{W}; this also appears subliminally 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
$\sB=\{P,1/r(a)\}$, 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(1/r(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~$k_C+2$, and each orbifold term is the inverse of
$\frac{1-t^a}{1-t}$ modulo $\frac{1-t^r}{1-t}$ written out as a
Laurent polynomial supported in the appropriate interval.
\end{exc}
\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[] My 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}
I 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''.
\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 $A_1=(1-t)^{n+1}A$ 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{figure}[ht]
\begin{verbatim}
function Qorb(r,LL,k)
L := [ Integers() | i : i in LL ]; // this 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
// Laurent polynomials properly
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{figure}
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 when the $a_i$
are coprime to $r$) and $l = [(k+n+1)/2]+\deg h$. (The $l$ just
translates the support of the Laurent polynomial. Now calculate
the hcf in the form
\begin{equation}
\hcf(A,t^l*B) =: H = \al * A+\be * t^l * B
\end{equation}
and return $\frac{t^l * be}{H * (1-t)^n * (1-t^r)}$. Since $\be$
is in the range $[0,r-2]$ it follows that $\be+l$ is in the range
\verb! [Floor((k+n+1)/2+h),Floor((k+n+1)/2+h)+r-2] !, as required.
\subsection{Orbifold in codim 0 and 1} \label{ssec!codim01}
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}.
\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
$
(3*t^5 - 2*t^4 + 3*t^3)/(t^8 - 3*t^7 + 3*t^6 - t^5 - t^3 + 3*t^2 - 3*t + 1)
8,3 ->
(8*t^7 - 12*t^6 + 17*t^5 - 12*t^4 + 8*t^3)
/
(t^10-4*t^9+7*t^8-8*t^7+8*t^6-8*t^5+8*t^4-8*t^3+7*t^2-4*t+1)
which after restoring the common factor (1+t) is
(8*t^8 - 4*t^7 + 5*t^6 + 5*t^5 - 4*t^4 + 8*t^3)/Denom[1,1,1,8]
Abstracting from that gives function called Nterm, of the form
(integral symm polynom supported in [3..r]) / [1,1,1,r]
with numerator Num uniquely determined by the condition
Num * (1-t^b)^2 (1-t^r-b)/(1-t)^3 == 1 + t^b mod (1-t^r)/(1-t)
(note the side-step k -> b is inverse of k mod r). It would be
jolly convenient to be able to calculate directly in the ring
QQ[t]/( (1-t^r)/(1-t) ). That is,
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;
The t^2*t^(r-2) is just a device for shifting the support into the
interval [3,..,r]. I can test this as much as I like:
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;
(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.
\end{verbatim}
\begin{thebibliography}{GW}
\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
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I think 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