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

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 ${\mathcal{M}}_{(x^{\prime },0,y)}$ 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.

32 pages. Final version with some simplifications and clarifications in the exposition. To appear in Invent. Math. (the final publication is available at springerlink.com)

References in corpus (3)

• Degenerations of Planar Linear Systems
• On the deformations of canonical double covers of minimal rational surfaces
• K3 double structures on Enriques surfaces and their smoothings

Cited by in corpus (3)

• On the deformations of canonical double covers of minimal rational surfaces
• An infinitesimal condition to deform a finite morphism to an embedding
• Positivity of line bundles on special blow ups of ${\mathbb{P}}^2$