Colloquium: A tale of two spaces: Hilbert schemes and Branch stacks

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 utilizing 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 called the Branch stack and conjectured its projectivity. I will describe my work on proving this conjecture, thereby enabling the Branch stack to be used in various applications.

Host: John Shareshian