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\title{Generic curves and non-coprime Catalans}

\author{Eugene Gorsky\thanks{Eugene Gorsky was partially supported by the NSF grants DMS-1760329 and DMS-1928930}\addressmark{1}, Mikhail Mazin\thanks{\href{mailto:mmazin@gmail.com}{mmazin@gmail.com}. Mikhail Mazin was partially supported by the Simons Collaboration grant 524324}\addressmark{2}, \and Alexei Oblomkov\thanks{Alexei Oblomkov was supported by the NSF grant DMS-1760373}\addressmark{3}}

\address{\addressmark{1}Department of Mathematics, UC Davis, Davis, California \\ \addressmark{2}Department of Mathematics, Kansas State University, Manhattan, Kansas\\
\addressmark{3}Department of Mathematics, UMass Amherst, Amherst, Massachusetts}

\received{\today}

\abstract{We compute the Poincar\'e polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents $(nd,md,md+1)$, and relate them to the combinatorics of $q,t$-Catalan numbers in the non-coprime case. We also confirm a conjecture of Cherednik and Danilenko for such curves.} 

\keywords{Compactified Jacobians, Rational Catalan Combinatorics, Dyck Paths}

\usepackage[backend=bibtex]{biblatex}
\addbibresource{generic_curves_fpsac.bib}

\begin{document}

\maketitle

\section{Introduction}

In this paper, we study the topology of compactified Jacobians of plane curve singularities. We focus on the case where the curve is reduced and locally irreducible (or {\em unibranched}), and it is known \cite{AIK,AK} that in this case compactified Jacobian is irreducible as well. 

Compactified Jacobians play an important role in modern geometric representation theory. First, they are closely related to Hilbert schemes of points on singular curves, singular fibers in the Hitchin fibration and  affine Springer fibers. In particular, counting points in the compactified Jacobians over finite fields is related to certain orbital integrals \cite{Laumon,KivTsai,Tsai}. Recent works \cite{GK,HKW,GKO} relate them to the representation theory of Coulomb branch algebras, defined by Braverman, Finkelberg and Nakajima \cite{BFN}. Second, a set of conjectures of the third author, Rasmussen and Shende \cite{OS,ORS} relates the homology of compactified Jacobians to the {\em Khovanov-Rozansky homology} of the corresponding knots and links. In particular, the conjectures imply that the homology of the compactified Jacobian is expected to be determined by the topology of the link or, in the unibranched case, by the collection of Puiseaux pairs of the singularity.

The progress in explicit computations of the homology of compactified Jacobians has been quite slow. For the quasi-homogeneous curves $C=\{x^m=y^n\}, \GCD(m,n)=1$, the homology was computed in many sources, starting with Lusztig and Smelt \cite{LS}. The key observation is that in this case $\overline{JC}$ admits a paving by affine cells. These cells and their dimensions have been given a number of combinatorial interpretations in \cite{GM1,GM2,GMV,Hikita}, where they were related to $q,t$-Catalan combinatorics. In \cite{OY1,OY2} the third author and Yun determined the ring structure on the homology in this case.

In a different direction, Piontkowski \cite{Piont} have computed the homology of compactified Jacobians for curves with one Puiseaux pair defined by the parameterization $(x,y)=(t^n,t^m+\ldots), \GCD(m,n)=1$. He showed that $\overline{JC}$ again admits an affine paving, and the combinatorics of cells depends only on $(m,n)$ and hence agrees with the quasi-homogeneous case $(x,y)=(t^n,t^m)$. Moreover, Piontkowski computed the cell decompositions for some curves with two Puiseaux exponents, where the combinatorics becomes rather subtle. In our main theorem, we vastly generalize the results of Piontkowski and prove the following.
\begin{theorem}
\label{thm: intro main}
Suppose $\GCD(m,n)=1$ and $d\ge 1$, consider the plane curve singularity $C$ defined by the parameterization
\begin{equation}
\label{eq: def generic}
(x(t),y(t))=(t^{nd},t^{md}+\lambda t^{{md}+1}+\ldots),\quad \lambda\neq 0
\end{equation}
Then:
\begin{enumerate}
\item[a)] The compactified Jacobian $\overline{JC}$ admits an affine paving where the cells are in bijection with Dyck paths $D$ in an $(nd)\times (md)$ rectangle. 
\item[b)] The Poincar\'e polynomial of $\overline{JC}$ is given by 
$$
P_{\overline{JC}}(t)=\sum_{D\in \Dyck(nd,md)}t^{2(\delta-\dinv(D))}
$$
where $\dinv$ is a certain statistics on Dyck paths defined in \cref{sec: combi}, and $\delta$ is the number of boxes weakly under the diagonal in an $(nd)\times (md)$ rectangle.
\item[c)] In particular, the Poincar\'e polynomial does not depend on $\lambda$ (as long as it is nonzero) or on the higher order terms in \eqref{eq: def generic}.
\end{enumerate}
\end{theorem}
We call the curves \eqref{eq: def generic} {\em generic curves}, since a generic curve with the first Puiseaux pair $(nd,md)$ has this form. The affine paving in the statement of \cref{thm: intro main} is obtained by intersecting the compactified Jacobian with the Schubert cell paving of the affine Grassmannian. Thus one can define a partial order on the strata such that  the boundary  of an affine cell  lies in the union of the cells with smaller indices in this order. Hence we conclude:
\begin{corollary}
For generic curves, the cohomology  $H^*(\overline{JC})$ is supported in even degrees and the weight filtration on  $H^*(\overline{JC})$ is pure in the sense of \cite{GKM,GKM2}.
\end{corollary}
Very recently, Kivinen and Tsai \cite{KivTsai} used completely different methods ($p$-adic harmonic analysis) to count points in arbitrary compactified Jacobians  over finite fields $\mathbb{F}_q$. They proved that the result is always a polynomial in $q$ and hence recovers the weight polynomial of  $\overline{JC}.$ Given the above purity result, for generic curves the Poincar\'e polynomial of $\overline{JC}$ agrees with the weight polynomial and our result agrees with theirs. 

As a corollary of \cref{thm: intro main}, we get that the Euler characteristic of $\overline{JC}$ is given by the number of Dyck paths in $(nd)\times (md)$ rectangle. For example, for the curve $C=(t^4,t^6+t^7)$ the Euler characteristic $\chi(\overline{JC})=23$ is equal to the number of Dyck paths in a $4\times 6$ rectangle, in agreement with \cite{Piont}. 

Next, we address the conjectures of Cherednik, Danilenko and Philipp \cite{CheD,CheP}, which proposed an expression for the Poincar\'e polynomials of compactified Jacobians in terms of certain matrix elements of certain operators in the double affine Hecke algebra, see \cref{sec: daha} for more details. We are able to prove this conjecture for generic curves:
\begin{theorem}
\label{thm: intro daha}
Consider the two-variable polynomial
\begin{equation}
\label{eq: def qtcat}
C_{nd,md}(q,t)=\sum_{D\in \Dyck(dn,dm)}q^{\area(D)}t^{\dinv(D)}
\end{equation}
where $\area(D)$ is the number of full boxes between a Dyck path $D$ and the diagonal. It satisfies the following properties:
\begin{enumerate}
\item[a)] It is symmetric in $q$ and $t$ : $C_{nd,md}(q,t)=C_{nd,md}(t,q)$
\item[b)] At $q=1$ it specializes to the Poincar\'e polynomial of $\overline{JC}$ (up to a linear change of the variable). 
\item[c)] It is given by the matrix element $(\gamma_{n,m}(e_d)(1),e_{nd})$ of the elliptic Hall algebra operator $\gamma_{n,m}(e_d)$.
\item[d)] It agrees with the Cherednik-Danilenko conjecture (\cref{conj: CD}).
\end{enumerate}
\end{theorem} 
Compositional Rational Shuffle Theorem \cite{Mellit} implies a different, manifestly symmetric in $q,t$, explicit formula for $C_{nd,md}(q,t),$ which implies part (a). Part (c) also follows from the Compositional Rational Shuffle Theorem, part (d) follows from (c). Part (b) is  a rephrasing of \cref{thm: intro main}. \cref{thm: intro daha}(a) allows us to give a simple formula for the Poincar\'e polynomial of the compactified Jacobian:
\begin{corollary}
The Poincar\'e polynomial of $\overline{JC}$ equals
$$
t^{2\delta}C_{nd,md}(1,t^{-2})=t^{2\delta}C_{nd,md}(t^{-2},1)=\sum_{D\in \Dyck(nd,md)}t^{2(\delta-\area(D))}.
$$
\end{corollary}
A more detailed version of the material presented in this extended abstract can be found in \cite{GMO}.
\section{Background}
\label{sec: background}
\subsection{Compactified Jacobians and semigroups}
Let $C$ be an irreducible (and reduced) plane curve singularity at $(0,0)$. We can parametrize $C$ as $(x(t),y(t))$, and write the local ring of functions on $C$ as 
$\cO_C=\C[[x(t),y(t)]]\subset \C[[t]].$ Given a function $f(t)\in \cO_C$, we can write 
$$
f(t)=\alpha_k t^{k}+\alpha_{k+1}t^{k+1}+\ldots,\ \alpha_k\neq 0
$$
and define the order of $f(t)$ by $\Ord f(t)=k$. The compactified Jacobian $\overline{JC}$ is defined as the moduli space of rank one torsion free sheaves on $C$ or, equivalently, $\cO_C$-submodules $M\subset \C[[t]]$. Note that $M$ is an $\cO_C$--submodule if and only if  $x(t)M\subset M,\ y(t)M\subset M$. Given such a subspace $M$, we define
$$
\Delta_M=\{\Ord\ f(t)\ :\ f(t)\in M\}\subset \Z_{\ge 0}.
$$
In particular, for $M=\cO_C$ we obtain the {\bf semigroup} of $C$:
$$
\Gamma=\Delta_{\cO_C}=\{\Ord\ f(t)\ :\ f(t)\in \cO_C\}.
$$
If $M$ is an $\cO_C$-submodule then $\Delta_M$ is a $\Gamma$-module: $\Delta_M+\Gamma\subset \Delta_M$. 
This motivates the following
\begin{definition}
Given a subset $\Delta\subset \Z_{\ge 0}$, we define the stratum in the compactified Jacobian:
$$
J_{\Delta}:=\{M\subset \C[[t]]: \cO_CM\subset M,\ \Delta_M=\Delta\}.
$$ 
\end{definition}
Clearly, $J_{\Delta}$ give a subdivision of $\overline{JC}$ and  $J_{\Delta}$ is empty if $\Delta$ is not $\Gamma$-invariant.  
As a warning to the reader, $J_{\Delta}$ could be empty even if $\Delta$ is $\Gamma$-invariant. 
\subsection{Generic curves}
 It is well known that any curve $C$ can be parametrized using Puiseaux expansion:
$$
x=t^{nd},\ y=t^{md}+\lambda t^{md+1}+\ldots,\ \GCD(m,n)=1.
$$
If $d=1$ then $C$ has one Puiseaux pair $(n,m)$ and its link is the $(n,m)$ torus knot. In this paper, we will be mostly interested in the case $d>1$.
\begin{definition}
Assume $d>1$. A curve $C$ is called {\em generic} if $\lambda \neq 0$. 
\end{definition}
It is easy to see that the definition of a generic curve is symmetric in $n$ and $m$. Indeed, we can choose the new parameter 
$$
\widetilde{t}=\sqrt[md]{y(t)}=t\sqrt[md]{1+\lambda t^{md+1}+\ldots}=t\left(1+\frac{\lambda}{md}t+\ldots\right),
$$
then 
$$
y=\widetilde{t}^{md},\ x=\widetilde{t}^{nd}-\frac{n\lambda }{md}\widetilde{t}^{nd+1}+\ldots.
$$
If $n=1$, then the curve has one Puiseaux pair $(d,md+1)$. Otherwise, it has two Puiseaux pairs $(nd,md)$ and $(d,md+1)$, which completely determine the topological type of the corresponding knot as a $(d,mnd+1)$-cable of the $(n,m)$ torus knot. 
\subsection{Invariant subsets}
\label{sec: comb gens}.
A subset $\Delta\subset \Z_{\ge 0}$ is called $0$-normalized if $0\in \Delta$. We will mostly consider $0$-normalized subsets, as any subset of $\Z_{\ge 0}$ can be shifted to a unique $0$-normalized one. We call $\Delta$ cofinite if $\Z_{\ge 0}\setminus \Delta$ is finite. For a cofinite subset $\Delta$ and $x\ge 0$ we write 
$$
\Gaps(x)=\Gaps_{\Delta}(x):=[x,+\infty)\setminus \Delta.
$$
\begin{definition}
We call $\Delta$ an $(nd)$-invariant subset if $nd+\Delta\subset \Delta$. A number $a$ is called an $(nd)$-generator of $\Delta$ if $a\in \Delta$ but $a-nd\notin \Delta$. 
\end{definition}
It is clear that for a cofinite $(nd)$-invariant subset $\Delta$ there is exactly one $(nd)$-generator in each remainder modulo $nd$. We will group them according to their remainders modulo $d$, so that the generators $a_{j,i},\ i=0,\ldots,n-1$ all have remainder $j$ modulo $d.$ We will further reorder $a_{j,i}$ so that $a_{j,i}+m\equiv a_{j,i+1}$ modulo $n.$ We write 
$$
\Delta_{j}=\bigcup_{i=0,\ldots,n-1} (a_{j,i}+dn\mathbb{Z}_{\ge 0}),\ \Delta=\bigcup_{j=0,\ldots, d} \Delta_j.
$$
We will call the integers $a_{j,i}+md$  the combinatorial syzygys of $\Delta$. It is clear that  in each remainder modulo $d$ there are $n$ such syzygys.
\section{Topology of compactified Jacobians}
\label{sec: geometry}
Throughout this section we fix a generic curve $C$ with parametrization $$(x(t),y(t))=(t^{nd},t^{md}+\lambda t^{md+1}+\ldots)$$ with $\lambda\neq 0$. We will use the notation $\cO_C=\C[[x(t),y(t)]]$ for the ring of functions on $C$.

We also fix a cofinite $(nd,md)$-invariant subset $\Delta\subset \Z_{\ge 0}$ with $(nd)$-generators $a_{j,i}$ as in \cref{sec: comb gens}.  We denote by $A=\{a_{j,i}\}$ the set of all $(nd)$-generators. The main goal of this section is to describe the stratum $J_{\Delta}$ in the compactified Jacobian $\overline{JC}$.
\subsection{Equations for $J_\Delta$}
\label{sec:equations-j_delta}
Consider an $\cO_C$-module $M\in  J_{\Delta}.$ 
\begin{lemma}
\label{lem: canonical}
Then for all $k\in\Delta$ there exist a unique canonical representative
\begin{equation*}
f_k=t^k+\sum_{l\in\Gaps(k)} f_{k;l-k} t^l \in M.
\end{equation*}
The canonical generators form a topological basis in $M$.
\end{lemma}
Vice versa, consider a collection of polynomials
\begin{equation*}
\left\{g_a=t^a+\sum_{l\in\Gaps(a)} g_{a;l-a} t^l| a\in A\right\}.
\end{equation*}
It will be convenient to consider the coefficients $g_{a;l-a}$ for $l\in\Gaps(a)$ as parameters. For $l\neq\Gaps(a)$ we use conventions
\[
g_{a;0}=1,\quad g_{a;l-a}=0 \mbox{ if } l\in \Delta\setminus\{a\}.
\]
Let $N$ be the $\cO_C$-submodule generated by the collection, and let $\widetilde{N}$ be the $\C[[t^{dn}]]$ submodule generated by the same collection. Then $N\in J_{\Delta}$ iff $N=\widetilde{N},$ or, equivalently,
\begin{equation*}
y(t)g_a \in\widetilde{N}\ \forall\ a\in A.
\end{equation*}
For every $a\in A$ let $s_{a+dm}$ be a polynomial in $t$ whose coefficients are polynomials in \(g_{a';r}\), \(a'\in A\), \(r\in \mathbb{Z}_{>0}\)  such that
\begin{equation*}
y(t)g_a-s_{a+dm}\in \widetilde{N}
\end{equation*}
and 
\begin{equation*}
  s_{a+dm}=\sum_{l\in\Gaps(a+dm)} s_{a+dm;l-dm-a} t^l.
\end{equation*}
The condition $y(t)g_a\in\widetilde{N}$ is equivalent to $s_{a+dm}=0$ for all \(a\in A\). Thus \(J_\Delta\) is a subset of the affine space \(\Gen=\C^{G(\Delta)}\) defined by \(E(\Delta)\) equations:
\[G(\Delta)=\sum_{i,j}|\Gaps(a_{j,i})|,\quad E(\Delta)=\sum_{i,j}|\Gaps(a_{j,i}+dm)|.\]
In particular, the coefficients \(g_{a_{j,i};x}\) for \(x\in \Gaps(a_{j,i})\) are natural coordinates on \(\Gen\). The equations are
\[s_{a_{j,i}+dm;x}(g)=0, \quad a_{j,i}+dm+x\notin\Delta.\]
\subsection{Admissibility}
\label{sec:comb-ag}
\begin{definition}
We call $x>0$ $j$-suspicious  if $a_{j,i}+md+x\notin \Delta$ for all $i$, and suspicious if it is $j$-suspicious for at least one remainder $j$.
\end{definition}
\begin{lemma}\label{lem:susp}
A number $x$ is $j$-suspicious  if and only if $\Delta_{j+x}\subset \Delta_j+md+nd+x$.
\end{lemma}
\begin{definition}
\label{def: admissible}
We call $\Delta$ admissible, if $1$ is not suspicious for $\Delta$.
\end{definition}
\begin{lemma}
\label{lem: admissible necessary}
Suppose that $J_\Delta\neq\emptyset.$ Then $\Delta$ is admissible.
\end{lemma}
\begin{proof}
Suppose $\Delta$ is not admissible, then there exists some $j$ such that $a_{j,i}+md+1\notin \Delta$ for all $i$. This implies $a_{j,i}+1\notin \Delta$ for all $i$, so we have canonical generators  $g_{a_{j,i}}=t^{a_{j,i}}+g_{a_{j,i};1}t^{a_{j,i}+1}+\ldots$ 
If $a_{j,i}+md=a_{j,i+1}+\alpha nd$ then we get  
\begin{align*}
t^{\alpha nd}g_{a_{j,i+1}}-y(t)g_{a_{j,i}}&=t^{\alpha nd}g_{a_{j,i+1}}-(t^{md}+\lambda t^{md+1}+\ldots)g_{a_{j,i}}\\\ &=(g_{a_{j,i+1};1}-g_{a_{j,i};1}-\lambda)t^{a_{j,i}+md+1}+\ldots
\end{align*}
hence
$g_{a_{j,i+1};1}-g_{a_{j,i};1}-\lambda=0$ for all $i$.  By adding these for all $i$ we get $n\lambda=0$, contradiction.
\end{proof}
\subsection{Paving by affine spaces}
\label{sec:paving-affine-spaces}
Recall that  \(\Gen\) has coordinates $g_{a_{j,i};x}$ where $a_{j,i}+x\notin \Delta$. Sometimes we will use the notation $g_{a_{j,i};x}$ for all $x$, assuming
\begin{equation}
\label{eq: g zero}
g_{a_{j,i};x}=0\quad \text{if}\ a_{j,i}+x\in \Delta.
\end{equation}
It is also convenient to consider $\Gen$ as a graded vector space
$\Gen=\bigoplus_{x=1}^\infty \Gen_x,$ where $\Gen_x$ is spanned by $g_{a_{j,i};x}$ with a fixed $x.$ There are coordinates on \(\Gen\) that are most suitable for study of equations \(s_k\) in the case \(\Delta\) is admissible. Let us define
\[g^{-}_{j,i;x}=g_{a_{j,i};x}-g_{a_{j,i+1};x}, \quad i=0,\dots,n-1.\] These functions  are linearly dependent, for example
  \(\sum_i g^-_{j,i;x}=0\). We choose a  subset of these  as follows:
\begin{itemize}
\item[(1)] If  \(x\) is \(j\)-suspicious, we define \(I(j;x)=\{0,\dots,n-2\}\). 
\item[(2)] If \(x\) is not \(j\)-suspicious, we can define \(I(j;x)\) to be a set of \(i\) such that \(a_{j,i}+dm+x\notin \Delta\). 
\end{itemize}
\begin{lemma}
\label{lem: linear independent}
The functions \(g^-_{j,i;x}\), \(i\in I(j;x)\) are linearly independent.
\end{lemma}
Finally, if there exists at least one integer $i$ such that $a_{j,i}+x\notin \Delta,$ then we set \[g^+_{j;x}=\sum_{i:a_{j,i}+x\notin \Delta}  g_{a_{j,i};x}.\] Thus if $I(j;x)\neq\emptyset$ then \(\{g^-_{j,i;x},g^+_{j;x}\}\), \(i\in I(j;x)\) are linearly independent linear coordinates on the space of generators \(\Gen_x\). For each \(j,x\) such that $I(j;x)\neq\emptyset$ let us fix a subset
  \(\bar{I}(j;x)\) such that \(\left\{g^-_{j,i;x},g^+_{j;x}, g_{a_{j,i'};x}\right\}\) \(i\in I(j;x)\),
  \(i'\in \bar{I}(j;x)\)  is a basis of linear coordinates on \(\Gen_x\). For $I(j;x)=\emptyset$ set $\bar{I}(j;x)=\{0,1,\dots,n-1\}.$
As we will see later the coordinates \(g_{a_{j,i'};x}\), \(i'\in \bar{I}(j;x)\) are not constrained by the equations for \(J_\Delta\), so we call them \textit{free variables}. On the rest of the coordinates  \(g^*_{*;*}\) we introduce a partial order generated as follows. Allowing $*$ to take any independent values,
  \[
  g^-_{*,*;x}<g^+_{*;x}<g^-_{*,*,x+1},
  \] 
coordinates $g^-_{*,*;x}$ are ordered in any arbitrary way, and
$$
g^+_{j+1;x}<g^+_{j;x} 
$$ 
(cyclic notation modulo $d$) for $j\neq d-x-1.$ 
\begin{lemma}\label{lemma:expansionl} For \(x\) such that \(a_{ji}+dm+x\notin \Delta\) we have
\begin{equation}\label{equation:gjix-}
s_{a_{j,i}+dm;x}=g_{j,i;x}^- + \mbox{polynomial in free variables and variables} < g^-_{j,i;x} 
\end{equation}
and if \(x\) is \(j\)-suspicious then
\begin{equation}\label{equation:gjx+}
\sum_{i=0}^{n-1}s_{a_{j,i}+dm;x}=\lambda g^+_{j;x-1}+\mbox{polynomial in free variables and variables} < g^+_{j;x-1} 
\end{equation}      
\end{lemma}
\begin{proposition}\label{prop:adm-tria}
  If \(\Delta\) is admissible then
  \[J_{\Delta}=\mathbb{C}^{\dim(\Delta)},\quad \dim(\Delta)=G(\Delta)-E(\Delta).\]
\end{proposition}
\begin{proof}
Recall that $J_\Delta$ is defined by the equations $s_{a_{j,i}+dm;x}=0$ for $j,i,x$ such that $a_{j,i}+dm+x\notin\Delta.$ We can modify this system of equations as follows. Whenever $x$ is $j$-suspicious, replace $s_{a_{j,n-1}+dm;x}=0$ by $\sum\limits_{i=0}^{n-1} s_{a_{j,i}+dm;x}=0.$ Clearly, the new system of equations is equivalent to the old one. Furthermore, according to \cref{lemma:expansionl}, the new system of equations expresses some of the elements of a basis of the space $\Gen$ in terms of the smaller variables with respect to the order $<.$ Therefore, one can use the equations to eliminate these variables one by one. Since $\dim\Gen=G(\Delta)$ and there are $E(\Delta)$ equations, we obtain the required result.
\end{proof}  
\section{Combinatorics}
\label{sec: combi}
\subsection{More on invariant subsets}
Let $\Delta$ be an $(nd,md)$-invariant subset. We call $b$ an $(md)$-cogenerator for $\Delta$ if $b\notin \Delta$ but $b+md\in \Delta$.
\begin{lemma}
\label{lem: dim gen cogen}
Let $\Delta$ be an admissible cofinite $(nd,md)$-invariant subset. The dimension of $J_{\Delta}$ equals to the number of pairs $(a,b)$ such that $a$ is an $(nd)$-generator of $\Delta$, $b$ is an $(md)$-cogenerator and $a<b$.
\end{lemma}
Let $\Theta$ be an $n,m$-invariant subset in $\mathbb{Z}.$ As before, $n,m$ are relatively prime. 
\begin{definition}
The {\em skeleton} $S$ of $\Theta$ is the union of the $n$-generators and $m$-cogenerators of $\Theta$.
\end{definition}
\subsection{Equivalence classes of $dn,dm$-invariant subsets}
Let us remind the definitions of the equivalence classes of invariant subsets from \cite{GMVDyck}.   Let $\Theta=\left\{\Theta_0^0,\ldots,\Theta_{d-1}^0\right\}$ be a collection of $0$-normalized $(n,m)$-invariant subsets. For every $(x_1,\ldots,x_{d-1})\in\mathbb{R}_{\ge 0}^{d-1}$ consider
\begin{equation*}
\Delta=\Delta(x_1,\ldots,x_{d-1}):=\bigcup_{k=0}^{d-1} (d\Theta_k^0 + x_k),
\end{equation*}
where $x_0=0.$ If all shift parameters $x_0,\ldots,x_{d-1}$ are integers with different remainders modulo $d,$ then $\Delta$ is a cofinite $0$-normalized $(nd,md)$-invariant subset. 

For each $\Theta_k^0$ let $S_k^0$ be its skeleton. Consider the space $\mathbb{R}_{\ge 0}^l$ of all possible shifts $x_1,\ldots,x_{d-1}.$ Consider the subset $\Sigma_{\Theta}\subset\mathbb{R}_{\ge 0}^{d-1}$ consisting of all shifts $x_1,\ldots,x_{d-1}$ for which there exists $i$ and $j$ such that 
\begin{equation*}
dS_i^0+x_i\cap dS_j^0+x_j\neq\emptyset.
\end{equation*}
Clearly, $\Sigma_{\Theta}$ is a hyperplane arrangement. We say that two $(nd,md)$-invariant subsets are equivalent if the corresponding shifts belong to the same connected component of the complement to $\Sigma_{\Theta}.$ One can show that every connected component of the complement to $\Sigma_{\Theta}$ contains at least one point corresponding to an $(dn,dm)$-invariant subset.
To summarize, in order to get all the equivalence classes of the cofinite $(dn,dm)$-invariant subsets, one should consider all possible $d$-tuples of $0$-normalized $(n,m)$-invariant subsets $\Theta^0_0,\ldots,\Theta^0_{d-1},$ and for each such $d$-tuple consider the set of connected components of the complement to $\Sigma_{\Theta}$ in the space of shifts. Furthermore, one should consider these connected components up to symmetry: if two of the subsets are equal $\Theta^0_i=\Theta^0_j$ then switching the corresponding shift coordinates $x_i$ and $x_j$ interchanges the connected components corresponding to the same equivalence class of $(dn,dm)$-invariant subsets.
\subsection{Admissible representatives}
Let $\Delta$ be a cofinite $0$-normalized $(dn,dm)$-invariant subset. In the spirit of the above notations, let $\Delta=\bigcup_{k} d\Theta_k^0 + x_k + k,$ where $(x_1,\ldots,x_{d-1})\in d\mathbb{Z}_{\ge 0}^{d-1}\subset\mathbb{R}_{\ge 0}^{d-1},$ $x_0=0,$ and $\Theta_0^0,\ldots,\Theta_{d-1}^0$ are $0$-normalized $(n,m)$-invariant subsets. Denote $\Theta_k:=\Theta_k^0+\frac{x_k}{d}$ for all $k\in\{0,\ldots, d-1\}.$  

In these notations we get that $\Delta$ is admissible if for every $k\in\{0,\ldots,d-1\}$ one has
\begin{equation*}
d\Theta_{k+1}+(k+1)\nsubseteq d\Theta_k+k+dn+dm+1,
\end{equation*}
or, equivalently,
\begin{equation*}
\Theta_{k+1}\nsubseteq \Theta_k+n+m.
\end{equation*}
The following is our key combinatorial result:
\begin{theorem}
\label{thm: unique admissible}
Every equivalence class contains a unique admissible representative.
\end{theorem}
Let $Inv(dm,dn)$ be the set of cofinite $0$-normalized $(dm,dn)$-invariant subsets. The following Theorem is the main result of \cite{GMVDyck}:
\begin{theorem}[\cite{GMVDyck}]
\label{thm: GMVDyck}
There exists a bijection $\mathcal{D}:\Inv(dm,dn)/{\sim} \to \Dyck(dm,dn),$ where $\sim$ is the equivalence relation defined above, and $\Dyck(dm,dn)$ is the set of $(dm,dn)$-Dyck paths. Furthermore, $\dim \Delta=\delta - \dinv(\mathcal{D}(\Delta))$
for all $\Delta\in \Inv(nd,md).$
\end{theorem}
\subsection{Proof of \cref{thm: intro main}}
We combine all of the above results to prove \cref{thm: intro main}. The compactified Jacobian $\overline{JC}$ is stratified into locally closed subsets  $J_{\Delta}$ and by \cref{prop:adm-tria} they are isomorphic to affine spaces of dimension $\dim(\Delta)$  if $\Delta$ is admissible and empty otherwise. Therefore the Poincar\'e polynomial has the form
$$
P(t)=\sum_{\Delta\ \mathrm{admissible}}t^{2\dim(\Delta)}.
$$
Next, we consider the infinite set $\Inv(nd,md)$ of all $(nd,md)$-invariant subsets in $\Z_{\ge 0}$ and the equivalence relation on it. By \cref{thm: unique admissible} in each equivalence class there is a unique admissible representative. Furthermore, by \cref{lem: dim gen cogen} the ``combinatorial dimension" $\dim(\Delta)$ depends only on the order of generators and cogenerators, and hence is constant on each equivalence class. Therefore we can write
$$
P(t)=\sum_{\Delta\in \Inv(nd,md)/\sim}t^{2\dim(\Delta)}.
$$
Finally, by \cref{thm: GMVDyck} there is a bijection between the equivalence classes and Dyck paths in $(nd)\times (md)$ rectangle, and the statistic $\dim(\Delta)$ on the former corresponds to the statistic $\codinv=\delta-\dinv$ on the latter, so
$$
P(t)=\sum_{D\in \Dyck(nd,md)}t^{2(\delta-\dinv(d))}.
$$
\section{Rational Shuffle Theorem and generic curves}
\label{sec: daha}
\subsection{Elliptic Hall algebra}
We briefly recall some notations for the elliptic Hall algebra \cite{BS}, and refer the reader to \cite{BGLX,GN,Mellit, Negut} for more precise statements and details. 

The elliptic Hall algebra $\cE$ is generated by elements $P_{kn,km}$ for all possible $(kn,km)$. The universal cover of the group $\mathrm{SL}(2,\Z)$ acts on $\cE$ by automorphisms. For coprime $m$ and $n$ we denote by $\gamma_{n,m}$ an element of $\mathrm{SL}(2,\Z)$ such that $\gamma_{n,m}(1,0)=(n,m)$. Then one gets $P_{kn,km}=\gamma_{n,m}(P_{k,0})$.

Let $\Lambda$ be the ring of symmetric functions in infinitely many variables. The elliptic Hall algebra $\cE$ acts on $\Lambda,$ and the multiplication operators by power sum symmetric functions $p_k$ correspond (up to a scalar) to the generators $P_{k,0}$ of $\cE$. Furthermore, $\cE$ is graded, and the grading is compatible with the grading on $\Lambda$. The generator $P_{kn,km}$ has degree $kn$.
\subsection{Cherednik-Danilenko conjecture}
\label{sec: CD}
Consider a Puiseaux expansion of plane curve singularity:
$$
y=b_1x^{\frac{m_1}{r_1}}+b_2x^{\frac{m_2}{r_1r_2}}+b_3x^{\frac{m_3}{r_1r_2r_3}}+\ldots,\ \quad b_i\neq 0.
$$
Here we assume $\GCD(m_i,r_1\cdots r_i)=1$.
The exponents are related to characteristic pairs $(r_i,s_i)$ by the equations $m_1=s_1,m_i=s_i+r_im_{i-1}\ (i>1)$. Given a sequence of characteristic pairs $(r_1,s_1),\ldots,(r_{\ell},s_{\ell})$, the authors of \cite{CheD} define a sequence of symmetric functions $f_{\ell+1},f_{\ell},\ldots,f_1$ by setting $f_{\ell+1}=p_1$ and $
f_{k}=\gamma_{r_\ell,s_\ell}(f_{k+1})(1),$ where $f_{k+1}$ is viewed as a multiplication operator on $\Lambda,$ and thus an element of $\cE.$
\begin{conjecture}[\cite{CheD}]
\label{conj: CD}
Let $f_1$ be the symmetric function of degree $r_1\cdots r_{\ell}$ obtained by the above procedure. The specialization of
$(f_1, e_{r_1\cdots r_\ell})$ at $t=1$ agrees with the Poincar\'e polynomial of the compactified Jacobian of an algebraic curve with characteristic pairs $(r_1,s_1),\ldots,(r_{\ell},s_{\ell})$. 
\end{conjecture}
In the case of generic curves \eqref{eq: def generic} we have
$$
y=x^{\frac{m}{n}}+\lambda x^{\frac{md+1}{nd}}+\ldots,\ m_1=m,\ r_1=n,\ m_2=md+1,\ r_2=d.
$$ 
This means that $s_1=m$ and $s_2=1$. To follow the above procedure, we first need to compute the operator
$\gamma_{r_2,s_2}(p_1)$ and the corresponding symmetric function $f_2$. By \cite[Corollary 6.5]{GN} we have $f_2=\gamma_{d,1}(p_1)(1)= P_{d,1}(1)=e_d.$ Therefore the next symmetric function is $f_1=\gamma_{n,m}(e_d)(1).$
Then $(f_1,e_{nd})=C_{nd,md}(q,t)$ follows from Rational Shuffle Theorem \cite{Mellit}. We conclude that for generic curves \cref{conj: CD} is true and follows from \cref{thm: intro main}.
\acknowledgements{We thank Francois Bergeron, Oscar Kivinen, Anton Mellit and Monica Vazirani for useful discussions.}
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