Equivariant smoothing of cusp singularities

TL;DR

This paper extends the theory of smoothing cusp singularities to an equivariant context and shows that certain smoothings can be achieved via locally complete intersection cusps, supporting the existence of related moduli stacks.

Contribution

It generalizes Looijenga's conjecture to equivariant settings and demonstrates that smoothings of cusp singularities can be induced by locally complete intersection cusps.

Findings

Smoothing of cusp singularities can be achieved equivariantly.

Any cusp singularity with a one-parameter smoothing can be smoothed via locally complete intersection cusps.

Supports the existence of the moduli stack of LCI covers over semi-log-canonical surfaces.

Abstract

We generalize Looijenga's conjecture for smoothing surface cusp singularities to the equivariant setting. Moreover, we prove that for any cusp singularity which admits a one-parameter smoothing, the smoothing can always be induced by smoothing of locally complete intersection cusps. The result provides evidence for the existence of the moduli stack of $\lci$ covers over semi-log-canonical surfaces.