SEMINAR:Smooth rational curves on (quasi-)polarized K3-surfaces

Guest: Alex Degtyarev, Bilkent University, 

Title: Smooth rational curves on (quasi-)polarized K3-surfaces (MATH, PHYS)

Date/Time: December 11, 2024, 13:40

Location:  https://sabanciuniv.zoom.us/j/96022533917, meeting ID: 960 2253 3917

Abstract: I will give a brief account of the current state of affairs in a very classical problem which, while going all the way back to Cayley, Solomon, and Schur, has become quite popular in the recent decades. It started from counting/bounding the number of lines on a smooth complex spatial surface. However, even though the original problem was closed long ago (essentially by Segre), the modern approaches and technology let us answer questions that we did not even dare to ask before and,as it is often the case, the more we learn the more we want to know, each result giving rise to a number of conjectures still waiting for their resolution. Thus, we can allow surfaces with singularities, consider other (quasi-)polarizations, count smooth rational curves of other degrees (conics, twisted cubics, etc.), and work over other fields of definition, including non-algebraically closed ones. Concentrating on the particular case of K3-surfaces (spatial quartics and plane sextic curves in the original setting), I will discuss the recent results and the tools used and work out a few examples, both classical and advanced.