O'Grady's Birational Maps via Wall-hitting

TL;DR

This paper interprets O'Grady's birational maps between moduli spaces of sheaves on elliptic K3 surfaces as wall-crossing transformations at specific semistable walls, linking geometric transformations to stability conditions.

Contribution

It provides a new interpretation of O'Grady's maps as wall-hitting transformations at totally semistable walls, and describes the first such wall for ideal sheaves on elliptic K3 surfaces.

Findings

O'Grady's birational maps are wall-crossing transformations at totally semistable walls.

Describes the first totally semistable wall for ideal sheaves of points on elliptic K3.

Remarks on the implications for Marian and Oprea's strange duality.

Abstract

We observe that O'Grady's birational maps between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at the so-called totally semistable walls, studied by Bayer and Macr\`i. As an ingredient to prove this observation, we describe the first totally semistable wall for ideal sheaves of $n$ points on the elliptic K3. We then use this observation to make a remark on Marian and Oprea's strange duality.