Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of \mathbb A ²

2024/07/10 by Piotr Oszer, Oszer, Piotr
Mathematics · #Graph theory and applications #Algebraic structures and combinatorial models #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2407.07993

Abstract

Using Hilbert-Burch matrices, we give an explicit description of the Białynicki-Birula cells on the Hilbert scheme of points on \mathbb A ² with isolated fixed points. If the fixed point locus is positive dimensional we obtain an étale rational map to the cell. We prove Conjecture 4.2 from arXiv:2309.06871 which we realize as a special case of our construction. We also show examples when the construction provides a rational étale map to the Hilbert scheme which is not contained in any Białynicki-Birula cell. Finally, we give an explicit description of the formal deformations of any ideal in the Hilbert scheme of points on the plane.

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