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On the Brill-Noether theory for K3 surfaces

Maxim Leyenson — 2012

Open Mathematics

Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which...

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman;  Rebecca Lehman;  Maxim Leyenson;  Mina Teicher — 2012

Journal of the European Mathematical Society

The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in ℙ 3 . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, B and E , we give a necessary and sufficient condition for B to be the branch curve of a surface X in ℙ N and E to be the image of the double curve of a ℙ 3 -model of X . In the classical Segre theory, a plane curve...

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