Joachim Jelisiejew

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 8 × 8 {8\times 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 Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quot d ⁡ ( 𝒪 𝔸 n ⊕ r ) {\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})} for d ≤ 7 {d\leq 7} and all r , n .