4-dimensional symplectic contractions

TL;DR

This paper explores the structure of four-dimensional symplectic contractions, showing they are Mori Dream Spaces connected by Mukai flops, and analyzes the movable cone and essential curves related to resolutions of surface Du Val singularities.

Contribution

It provides a detailed study of the movable cone, essential curves, and their relation to resolutions of surface Du Val singularities in four-dimensional symplectic contractions.

Findings

Movable cone divided into nef chambers related to different resolutions

Essential curves determine the faces of the movable cone

Schemes parametrizing minimal essential curves are resolutions of surface Du Val singularities

Abstract

Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes parametrizing minimal essential curves and show that they are resolutions, possibly non-minimal, of surface Du Val singularities. Some examples, with an exhaustive description, are provided.