Orthogonal exceptional collections from $\mathbb{Q}$-Gorenstein degeneration of surfaces

TL;DR

This paper constructs orthogonal exceptional vector bundles on surfaces degenerating to cyclic quotient singularities, expanding the understanding of derived categories in algebraic geometry.

Contribution

It introduces a method to produce orthogonal exceptional bundles from $Q$-Gorenstein degenerations to specific cyclic quotient singularities.

Findings

Constructed $d$ exceptional vector bundles of rank $n$

Bundles are orthogonal to each other

Applicable under certain technical assumptions

Abstract

We consider a surface that admits a $\mathbb{Q}$-Gorenstein degeneration to a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$. Under several technical assumptions, we construct $d$ exceptional vector bundles of rank $n$ which are orthogonal to each other.