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

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