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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 of8  ×  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 ofQuot  d   ⁡   (        n    ⊕  r    ){\operatorname{Quot}_{d}(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})}    ford  ≤  7{d\leq 7}    and all  r ,  n .

Components and singularities of Quot schemes and varieties of commuting matrices