Secant v. Cactus

Quot schemes and varieties of commuting matrices

11 mai 2023, 11:00

1h

Building 1R3, Amphitheater Schwartz (Institut de Mathématiques de Toulouse)

Orateur

Klemen Sivic (University of Ljubljana)

Description

Let $C_n(M_d)$ denote the affine variety of all $n$-tuples of commuting $d\times d$ matrices. The ADHM construction relates these varieties to Quot schemes, and in particular to Hilbert schemes. On the more applied side, varieties $C_n(M_d)$ are directly connected to the question whether a tensor has minimal border rank. Although $C_n(M_d)$ is usually reducible for $n>2$ and $d>3$, very few irreducible components are known. In the talk we classify irreducible components for small $d$ and all $n$. Moreover, we show that $C_n(M_d)$, viewed as a scheme defined by the quadratic commutativity relations, has generically nonreduced components whenever $d\ge 8$ and $n\ge 4$, while it is generically reduced for $d\le 7$. Our results give the corresponding results for Quot schemes of points. In particular, the Quot scheme parametrizing degree 8 quotients of a free module of rank 4 over polynomial ring in 4 variables has a generically nonreduced component.
This is joint work with Joachim Jelisiejew.