Deformation of canonical morphisms and the moduli of surfaces of general type

TL;DR

This paper develops deformation theory for finite maps to construct and analyze surfaces of general type, revealing new components in their moduli spaces and criteria for deforming finite maps to embeddings.

Contribution

It introduces a criterion for deforming finite maps into embeddings and applies it to construct new canonical surfaces with varied invariants and moduli components.

Findings

Constructed new simple canonical surfaces with different invariants.

Identified new components in the moduli space of surfaces of general type.

Established a criterion for deforming finite maps to one-to-one maps.

Abstract

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one--to--one map. We use this criterion to construct new simple canonical surfaces with different $c_1^2$ and $\chi$. Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces $\mathcal M_{(x',0,y)}$ having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism.