PUBLIC VIVA-VOCE NOTIFICATION
3:30 pm,Seminar Hall
Seshadri constants on algebraic surfaces
Praveen Kumar Roy
Chennai Mathematical Institute.
11-12-19

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Abstract

For an ample line bundle L on a projective variety X, J.-P. Demailly defined an invariant of L which studies the local geometry/positivity of L around a point x in X. He used Seshadri's criterion of ampleness for a line bundle to define the Seshadri constant at a point. Since then this area attracted significant attention of many researchers exploring this area. Computing and bounding Seshadri constants of ample line bundles on varieties became an active area of research.

In this thesis we worked on computing and bounding the Seshadri constants on hyperelliptic surfaces. Hyperelliptic surfaces are nonsingular minimal surfaces of Kodaira dimension 0 and irregularity 1. They are realised as finite group quotients of a product of two elliptic curves. Their classification is well understood. We show that for all ample line bundles on hyperelliptic surfaces, global Seshadri constants are rational (except for one case in which we give a partial answer) and we compute them precisely in some cases. We also give some bounds extending the already known bounds. We extend the work of Lucja Farnik which motivated our study of Seshadri constants on hyperelliptic surfaces.

We also discuss surfaces of general type in this thesis. Surfaces of general type are surfaces with maximum Kodaira dimension, i.e., 2. Motivated by results of Thomas Bauer and Tomasz Szemberg, we prove a result about the multi-point Seshadri constant on a surface of general type. Next, we consider a surface of general type of the form C x C. Here C is a general member of the moduli of smooth curves of genus g, where g is greater than 2. In this case, we prove that the global Seshadri constant of an ample line bundle, under some restrictions, is rational. We also compute single point Seshadri constant precisely in this case for the canonical line bundle.