Speaker: Ritvik Ramkumar
Cornell University
Will give a Colloquium entitled:
A tale of two spaces: Hilbert schemes and Branch stacks
Abstract: Varieties are the solution sets of polynomial equations. A distinctive feature of algebraic geometry is that certain collections of varieties are often themselves a variety. A fundamental example is the Hilbert scheme, constructed by Grothendieck in the 1960s. When the varieties are collections of points on a surface, the associated Hilbert scheme is smooth and has found significant applications beyond algebraic geometry. However, when the surface is replaced by a threefold, the Hilbert scheme becomes singular, and its structure remains largely unexplored. I will outline my research program for studying these singularities introducing methods from algebra, geometry, and combinatorics. I will also highlight how the tools developed can be applied to other problems. To study higher-dimensional varieties, such as collections of curves, Alexeev and Knutson proposed a better alternative to the Hilbert scheme, known as the Branch stack. They conjectured its projectivity, a property that would make it as versatile as the Hilbert scheme. I will describe my work on proving this conjecture, thereby enabling the Branch stack to be used in various applications.
Date: 01-23-2025
Time: 4:00 pm
Location: 231 Hayes-Healy
Zoom URL: notredame.zoom.us/j/
Originally published at math.nd.edu.