Holomorphic pre-symplectic form on the nested Hilbert scheme $\mbox{Hilb}^{3,4}(\mathbb{C}^2)$

TL;DR

This paper explores the geometric structure of a specific moduli space of sheaves on the projective plane, focusing on a holomorphic pre-symplectic form and its degenerate points, using an adapted ADHM construction.

Contribution

It introduces a novel analysis of the pre-hyperk"ahler structure and classifies degenerate points of the holomorphic pre-symplectic form on the moduli space.

Findings

Identification of degenerate points of the pre-symplectic form

Application of ADHM construction to framed sheaves

Categorization of geometric degeneracies

Abstract

We regard the pre-hyperk\"ahler structure on the moduli space of framed flags of sheaves $\mathcal{F}(1,3,1)$ on the projective plane $\mathbb{P}^2$ via an adaptation of the ADHM construction of framed sheaves. Then, we study and categorize the degenerate points of the holomorphic pre-symplectic form presented in the moduli space.