Rationality of cohomological descendent series for Quot schemes on surfaces with $p_g=0$

TL;DR

This paper proves the rationality of cohomological descendent series for Quot schemes on surfaces with zero geometric genus, using wall-crossing and local curve factorization techniques.

Contribution

It establishes the rationality of these series in the case of surfaces with $p_g=0$, extending previous results to a broader class of surfaces.

Findings

Series are rational for $p_g=0$, $eta eq 0$, $N>1$ surfaces.

Uses a wall-crossing recursion of Pandharipande-Thomas type.

Reduces problem to local curve factors and vanishing results.

Abstract

For a smooth projective surface $S$, Johnson, Oprea, and Pandharipande defined cohomological descendent generating series for the Quot schemes of rank-$0$ quotients of $\mathcal{O}_S^{\oplus N}$. We prove that these series are rational in the remaining surface case $p_g(S)=0$, $\beta\neq 0$, and $N>1$. The proof uses a fixed-source one-parameter wall-crossing recursion of Pandharipande-Thomas type, a factorization through pure Quot by two explicit zero-dimensional correction operators, a support-flat reduction of the first correction to relative Quot theory on curves, rationality of the resulting smooth and singular local curve factors, and a local $K$-theoretic vanishing that collapses the second correction to the universal punctual smooth-surface factor.