Surface quotient singularities and bigness of the cotangent bundle: Part II

TL;DR

This paper develops a criterion for the bigness of cotangent bundles on resolutions of orbifold surfaces, applies it to hypersurfaces and cyclic covers, and derives formulas for invariants of singularities.

Contribution

It introduces the CMS criterion for big cotangent bundles, provides formulas for $A_n$-singularity invariants, and extends symmetric differential results.

Findings

Formulas for $A_n$-singularity invariants

Criteria for big cotangent bundles on surfaces

Extension results for symmetric differentials

Abstract

In two parts, we present a bigness criterion for the cotangent bundle of resolutions of orbifold surfaces of general type. As a corollary, we obtain the \textit{canonical model singularities} (CMS) criterion that can be applied to determine when a birational class of surfaces has smooth representatives with big cotangent bundle and compare it with other known criteria. We then apply this criterion to the problem of finding the minimal degrees $d$ for which the deformation equivalence class of a smooth hypersurface of degree d in $\PP^3$ has a representative with big cotangent bundle; applications to the minimal resolutions of cyclic covers of $\PP^2$ branched along line arrangements in general position are also obtained. The CMS criterion involves invariants of canonical singularities whose values were unknown. In this part of the work, we describe a method of finding these invariants and obtain formulas for $A_n$-singularities. We also use our approach to derive several extension results for symmetric differentials on the complement of the exceptional locus $E$ of the minimal resolution of an $A_n$-singularity; namely, we characterize the precise extent to which the poles along $E$ of symmetric differentials on the complement are milder than logarithmic poles.