The geometry and singularities of the Bilinear scheme

TL;DR

This paper investigates the geometry and singularities of the Bilinear scheme, relating it to Quot schemes, computing tangent spaces, and analyzing irreducibility and degeneracy loci using tensor theory.

Contribution

It establishes the representability of the Bilinear functor as a closed subscheme of Quot schemes and analyzes its geometric properties and reducibility.

Findings

The Bilinear scheme can be realized as a closed subscheme of a product of Quot schemes.

The main irreducible component corresponds to tuples of points.

The scheme is reducible for all n when r_i ≥ d ≥ 3.

Abstract

The goal is to study the geometry of the Bilinear scheme $\mathrm{Bilin}_{d_1,d_2,d_3}^{r_1, r_2}(\mathbb{A}^n)$ introduced by Joachim Jelisiejew. This functor can be viewed as a generalization of the Quot scheme, giving the moduli space of bilinear maps of locally free modules. We describe the relation to the Quot scheme by proving that the Bilinear functor can be realized as a closed subfunctor of a product of Quot schemes, hence the Bilinear functor is representable by a closed subscheme of the product of Quot schemes. We use this result to compute the tangent space to the Bilinear scheme representing $\mathrm{Bilin}_{d_1,d_2,d_3}^{r_1, r_2}(\mathbb{A}^n)$. We define two types of loci: the locus corresponding to tuples of points, and the totally degenerate locus. The first locus gives the main irreducible component of the Bilinear scheme. We use the theory of minimal border rank tensors and secant varieties, and find that $\mathrm{Bilin}_{d_1,d_2,d_3}^{r_1, r_2}(\mathbb{A}^n)$ is reducible for all $n$ whenever $r_i \geq d \geq 3$. We describe the $\Bbbk$-points of $\mathrm{Bilin}_{2,2,2}^{2, 2}(\mathbb{A}^1)$ in detail.