A Superficial Working Guide to Deformations and Moduli

TL;DR

This paper provides an introductory guide to deformations and moduli of algebraic surfaces, introduces new results on local homeomorphisms between Kuranishi and Teichmueller spaces, and surveys recent findings on deformation actions of groups.

Contribution

It offers new insights into the relationship between deformations of surfaces and their canonical models, including novel results on Albanese maps and group actions.

Findings

Def(S) maps properly onto Def(X) for surfaces and their canonical models.

Def(S,G) does not map properly onto Def(X,G) for group actions.

Connected components of Def(S) for tertiary Burniat surfaces map to locally closed sets.

Abstract

This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.