EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds

TL;DR

This paper demonstrates that certain EPW varieties related to Gushel-Mukai surfaces and threefolds can be interpreted as moduli spaces of semistable objects in their derived categories, linking geometric and categorical perspectives.

Contribution

It establishes the realization of EPW varieties as moduli spaces within derived categories for both Gushel-Mukai surfaces and special threefolds, extending categorical understanding.

Findings

EPW sextics associated with GM surfaces are moduli spaces of semistable objects.

EPW surfaces related to GM threefolds are also realized as moduli spaces.

Refinement of Bayer and Perry's statement on GM threefolds with equivalent Kuznetsov components.

Abstract

We show that the double dual EPW sextic and double EPW sextic associated with a strongly smooth Gushel-Mukai surface can be realized as moduli spaces of semistable objects with respect to a stability condition on its bounded derived category. Also, we observe that the double dual EPW surface and double EPW surface associated with a special Gushel--Mukai threefold can be realized as moduli spaces of semistable objects on its Kuznetsov component. As an application, we refine a statement of Bayer and Perry about Gushel--Mukai threefolds with equivalent Kuznetsov components for the special ones.