You are cordially invited to the seminars organized by the Department of Mathematics.
Speaker: Neelarnab Raha (Penn State)
“Brill-Noether theory for vector bundles on surfaces”
Abstract: The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty, and these loci can be reducible and of larger than the expected dimension.
Let $E$ be a semistable vector bundle on the projective plane. We give an upper bound $\beta_{r,\mu}$ for $h^0(E)$ in terms of the rank $r$ and the slope $\mu$ of $E$. We show that the bound is achieved precisely when $E$ is a Steiner bundle. We classify those $E$ for which $h^0(E)$ is close to $\beta_{r,\mu}$. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci with $h^0(E)$ in this range. When they are proper subvarieties, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension. This is joint work with Professors Izzet Coskun and Jack Huizenga.
We also have a similar bound on the smooth quadric surface, although it is more subtle depending on how balanced the first Chern class is, and what global generation properties the vector bundle possesses.
Date: November 14, Friday
Time: 6:00 P.M. (Turkey)
Place: Zoom
To request the event link, please send a message to turker.ozsari@bilkent.edu.tr