3:30-4:45pm, Seminar Hall
Deformation of Canonical morphisms and Moduli spaces (with F. J. Gallego and M. Gonzalez)
Purnaprajna P. Bangere
University of Kansas.
05-07-17

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Abstract

In this talk we will deal with 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 show a criterion that determines when a finite map can be deformed to a one--to--one map. We use this general result that holds for all dimensions, to construct new surfaces of general type with birational canonical map, for different $c_1^2$ and $\chi$ (the canonical map of the surfaces we construct is in fact a finite, birational morphism), addressing a question of Enriques posed in 1944. All known families until now were complete intersections or divisors in three folds.

Our results enable us to describe some new components of the moduli of surfaces of general type. We 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, the so called hyperelliptic components. We also prove two analogues of these results for varieties of general type in all dimensions.