Recent developments on Gromov-Witten theory of Hilbert schemes of points on certain surfaces

TL;DR

This paper surveys recent advances in Gromov-Witten invariants for Hilbert schemes of points on elliptic and minimal surfaces, emphasizing computational progress in lower genus cases and discussing key conjectures.

Contribution

It provides a comprehensive overview of recent developments, focusing on computational methods and conjectures in the Gromov-Witten theory of Hilbert schemes on specific surfaces.

Findings

Progress in computing Gromov-Witten invariants for lower genus cases

Formulation of important conjectures in the field

Compilation of current knowledge and research directions

Abstract

In This paper, we survey recent progress on the theory of Gromov- Witten invariants on Hilbert schemes of points mainly on elliptic surfaces and simply connected minimal surface of general type. In particular, we focus on the aspects of computational progress that has been done in the cases of lower genus. Then, we discuss some important conjectures that have been proposed and gather all available information and progress toward answering them.