Components and singularities of Quot schemes and varieties of commuting matrices

@article{Jelisiejew2022ComponentsAS,
  title={Components and singularities of Quot schemes and varieties of commuting matrices},
  author={Joachim Jelisiejew and Klemen {\vS}ivic},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={2022},
  volume={2022},
  pages={129 - 187}
}
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 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… 

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