Components and singularities of Quot schemes and varieties of commuting matrices
Abstract
We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of size 8 matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and BiałynickiBirula decompositions to this setup. We include a thorough review of our methods to make the paper selfcontained and accessible to both algebraic and linearalgebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of $Quot_d(O_{\mathbb{A}^n}^{\oplus r})$ for $d\leq 7$ and all $r$, $n$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.13137
 Bibcode:
 2021arXiv210613137J
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 Mathematics  Representation Theory;
 14C05;
 15A27;
 13E10
 EPrint:
 v4: final, minor corrections v3: minor v2: light corrections to initial submission. Comments still welcome v1: Comments most welcome! Initial submission: slight changes probably will occur. 52 pages