On jet schemes of determinantal varieties

2025/06/28 by Chen, Yifan, Zuo, Huaiqing · 1 citation
#14M12 14E18 05E40 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.22898

Abstract

Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the k-th order jet scheme of the determinantal variety defined by r-minors in an m × n matrix as \mathscrL^m,n_r,k. For the special case where m, n, and r are equal, and m and r are 3 while k is 1, we establish a correspondence between the defining ideals of \mathscrL^m,n_r,k and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of \mathscrL^m,n_r,k accordingly. Moreover, for general \mathscrL^m,n_r,k, \cite12 provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components. Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series.

Citations

• Algebraization and Tannaka duality
• Initial Complex Associated to a Jet Scheme of a Determinantal Variety
• Jet schemes, arc spaces and the Nash problem
• Jet schemes and singularities
• Groebner bases and determinantal ideals
• Determinantal Varieties Over Truncated Polynomial Rings
• Singularities of Pairs via Jet Schemes

Cited by

• Hilbert series of second order jets of determinantal varieties

Related

• Hilbert series of second order jets of determinantal varieties
• Jet schemes of determinantal varieties
• Initial Complex Associated to a Jet Scheme of a Determinantal Variety
• Minimal log discrepancies of determinantal varieties via jet schemes
• Determinantal Varieties Over Truncated Polynomial Rings