Exceptional bundles associated to degenerations of surfaces

TL;DR

This paper constructs a correspondence between degenerations of surfaces and exceptional vector bundles, revealing new links between surface singularities, moduli space boundaries, and vector bundle theory.

Contribution

It introduces a novel construction of exceptional bundles from surface degenerations, establishing bijections and connections in special cases.

Findings

Bijective correspondence for Y = projective plane

Connection between moduli space boundary components and exceptional bundles

Construction applicable to surfaces with specific H^{2,0} and H^1 conditions

Abstract

In 1981 J. Wahl described smoothings of surface quotient singularities with no vanishing cycles. Given a smoothing of a projective surface X of this type, we construct an associated exceptional vector bundle on the nearby fiber Y in the case H^{2,0}(Y)=H^1(Y)=0. If Y is the projective plane we show that our construction establishes a bijective correspondence between the possible degenerate surfaces X and exceptional bundles on Y modulo dualizing and tensoring by line bundles. If Y is of general type then our construction establishes a connection between components of the boundary of the moduli space of surfaces deformation equivalent to Y and exceptional bundles on Y.