Algebraic and Arithmetic Geometry Seminar

# The Hilbert scheme of infinite affine space

## by Burt Totaro (UCLA)

Europe/Rome
I will discuss the Hilbert scheme of d points in affine $n$-space, with some examples. This space has many irreducible components for $n$ at least $3$ and is poorly understood. Nonetheless, in the limit where n goes to infinity, we show that the Hilbert scheme of d points in infinite affine space has a very simple homotopy type. In fact, it has the $\mathbb{A}^1$-homotopy type of the infinite Grassmannian $\mathsf{BGL}(d-1)$. Many questions remain. (Joint with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson.)