Mathematics – Algebraic Geometry
Scientific paper
2010-01-07
Mathematics
Algebraic Geometry
32 pages. Final version with some simplifications and clarifications in the exposition. To appear in Invent. Math. (the final
Scientific paper
10.1007/s00222-010-0257-8
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.
Gallego Francisco Javier
Gonzalez Martinez M.
Purnaprajna Bangere P.
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