|
Public viva-voce Notification Date: Thursday, 17 July 2025 Time: 11:00 AM Venue: Seminar Hall (Hybrid mode) Linear systems and Seshadri constants on blow ups of ruled surfaces Cyril J Jacob Chennai Mathematical Institute. 17-07-25 Abstract In this talk, we study ruled surfaces over the projective line (Hirzebruch surfaces) and the projective plane. First, we will discuss the interpolation problem for curves in the projective plane. The well-known Segre-Harbourne-Gimigliano-Hirschowitz (SHGH) conjecture gives an expected answer to the interpolation problem in the projective plane. We will state some equivalent versions and a few implications of the SHGH conjecture, including the famous Nagata conjecture. After discussing these in the context of the projective plane, we will present analogous conjectures for Hirzebruch surfaces. We will show some positive results in this direction. In the second part of the talk, we focus on Seshadri constants, which measure the local positivity of an ample line bundle on a projective variety. The Seshadri constant was introduced by J.-P. Demailly in order to study the Fujita conjecture, and the definition is motivated by a characterisation of ampleness of line bundles given by C. S. Seshadri. Three of the most important questions in the study of Seshadri constants involve their computation, finding lower bounds, and exploring their potential irrationality. In this talk, we examine these questions within the context of blow ups of the projective plane as well as in blow ups of Hirzebruch surfaces. We show some lower bounds for Seshadri constants on blow ups of the projective plane under certain assumptions. Also, we compute exact values of Seshadri constants on some blow ups of Hirzebruch surfaces, and show that there exists an ample line bundle on a blow up of Hirzebruch surfaces at sufficiently many points with an irrational Seshadri constant, assuming a conjecture is true. All are invited to attend the viva-voce examination.
|
