Components of Hilbert Schemes of low degree and smoothable algebras

2025/12/10 by Gałązka, Maciej, Keneshlou, Hanieh, Šivic, Klemen
#13E10 #14C05 #14N07 #15A27 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.09428

Abstract

In this article, we describe the irreducible components of the Hilbert scheme of d points on \mathbbAⁿ for d=9,10. The main techniques we use are the variety of commuting matrices and analyzing loci of local algebras with a specific Hilbert function. We further prove that any finite local algebra of degrees 9,10 and the socle dimension 2 is smoothable. As the main consequence, we establish the equality of the cactus Grassmann and the secant Grassmann variety in the corresponding cases.

Citations

• Grassmann cactus variety and socle dimension
• New components of Hilbert schemes of points and 2-step ideals
• The Hilbert scheme of points on a threefold, I
• Open problems in deformations of Artinian algebras, Hilbert schemes and around
• On construction of k-regular maps to Grassmannians via algebras of socle dimension two
• Components and singularities of Quot schemes and varieties of commuting matrices
• Distinguishing secant from cactus varieties
• The Hilbert scheme of 11 points in A³ is irreducible
• Apolarity and direct sum decomposability of polynomials
• Secant varieties to high degree Veronese reembeddings, catalecticant\n matrices and smoothable Gorenstein schemes
• Lectures on Instanton Counting

Cited by

• Grassmann cactus variety and socle dimension

Related

• Small elementary components of Hilbert schemes of points
• The Hilbert scheme of 11 points in A³ is irreducible
• Generically nonreduced components of Hilbert schemes on fourfolds
• Local Equations for Hilbert Schemes of Points
• The Hilbert scheme of points and its link with border basis