Determinantal Varieties Research Articles

On the topology of determinantal links

AbstractWe study sections of the generic determinantal varieties by generic hyperplanes of various codimensions , the polar multiplicities of these sections, and the cohomology of their real and complex links. Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a few further special cases. Our results now allow to compute all the cohomology with integer coefficients below the middle degree and the middle Betti number for any determinantal smoothing.

Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence

We generalize Aganagic-Okounkov's theory of elliptic stable envelopes, and its physical realization in Dedushenko-Nekrasov's and Bullimore-Zhang's works, to certain varieties without holomorphic symplectic structure or polarization. These classes of varieties include, in particular, classical Higgs branches of 3d $\mathcal N=2$ quiver gauge theories. The Bethe/gauge correspondence relates such a gauge theory to an anisotropic/elliptic superspin chain, and the stable envelopes compute the $R$-matrix that solves the dynamical Yang-Baxter equation (dYBE) for this spin chain. As an illustrative example, we solve the dYBE for the elliptic $\mathfrak{sl}(1|1)$ spin chain with fundamental representations using the corresponding 3d $\mathcal N=2$ SQCD whose classical Higgs branch is the Lascoux resolution of a determinantal variety. Certain Janus partition functions of this theory on $I \times \mathbb E$ for an interval $I$ and an elliptic curve $\mathbb E$ compute the elliptic stable envelopes, and in turn the geometric elliptic $R$-matrix, of the anisotropic $\mathfrak{sl}(1|1)$ spin chain. Furthermore, we consider the 2d and 1d reductions of elliptic stable envelopes and the $R$-matrix. The reduction to 2d gives the K-theoretic stable envelopes and the trigonometric $R$-matrix, and a further reduction to 1d produces the cohomological stable envelopes and the rational $R$-matrix. The latter recovers Rimányi-Rozansky's results that appeared recently in the mathematical literature.

B-brane Transport and Grade Restriction Rule for Determinantal Varieties

B-brane Transport and Grade Restriction Rule for Determinantal Varieties

RETRACTED ARTICLE: Linear sections of determinantal varieties

Denote Mmn(k) to be the set of all m n matrices over k, and PMmn(k) to be its affine space. Let Dm,n,r PMmn(k) be the subvariety consisting of the projective classes of all m n matrices with rank less than or equal to r. We study the minimal integer k > 0 such that for any linear subspace L of dimension k in PMmn(k), Dm,n,r L = . We get some partial results about this problem. Specially, when m = n = r + 1, we solve this problem for k = R, C and Q.

Linear sections of determinantal varieties

Denote Mmn(k) to be the set of all m n matrices over k, and PMmn(k) to be its affine space. Let Dm,n,r PMmn(k) be the subvariety consisting of the projective classes of all m n matrices with rank less than or equal to r. We study the minimal integer k > 0 such that for any linear subspace L of dimension k in PMmn(k), Dm,n,r L = . We get some partial results about this problem. Specially, when m = n = r + 1, we solve this problem for k = R, C and Q.

Gröbner bases for bipartite determinantal ideals

Nakajima’s graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals. We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia–Guccione–Guccione. As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics.

Corrigendum to “Tropical Positivity and Determinantal Varieties”

Corrigendum to “Tropical Positivity and Determinantal Varieties”

Linear codes associated to determinantal varieties in the space of hermitian matrices

Linear codes associated to determinantal varieties in the space of hermitian matrices

On area-minimizing Pfaffian varieties

There are two significant families of minimal real matrix varieties: determinantal varieties and skew-symmetric determinantal varieties, the later ones are also known as Pfaffian varieties. In 1999, Kerckhove and Lawlor [26] proved that determinantal varieties are area-minimizing except for two families. In this paper we prove that all Pfaffian varieties are area-minimizing with the exception of Pfaffian hypersurfaces.

On nonsingular bilinear maps over different fields

The investigation of nonsingular bilinear forms originates from the classification of division algebras over the real number field. Building upon this foundation, researchers have delved into the study of nonsingular bilinear forms over real number fields, leading to significant results such as Hopfs theorem. However, the interest in understanding nonsingular bilinear forms extends beyond real number fields, prompting a desire to explore other fields as well. When it comes to algebraically closed fields, the theorem becomes well-understood, with essence captured by the Hopf-Smith theorem. Inspired by these established studies, we are motivated to further the comprehension of nonsingular bilinear forms over arbitrary fields. Given a field , we study in this article numerical constraints on for the existence of nonsingular bilinear maps for not only algebraically closed fields and the real number field but also the rational number field and finite fields. We reach the final conclusion mainly through algebro-geometric techniques and the use of determinantal varieties. We reprove a result of HopfSmith which states that the minimal possible value of is when is an algebraically closed field. When is the real number field, we prove that under a combinatorial condition, the minimal possible value of is still . We also show that when is the rational number field or a finite field, the minimal possible value of is .

On intersections of symmetric determinantal varieties and theta characteristics of canonical curves

From a block-diagonal (n+1)×(m+1)×(m+1) tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces X in n-dimensional projective space, and an intersection of quadrics C in m-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of X in terms of C. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of Cayley.

Determinantal Varieties From Point Configurations on Hypersurfaces

Abstract We consider the scheme $X_{r,d,n}$ parameterizing $n$ ordered points in projective space $\mathbb {P}^{r}$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure, and we prove that it is irreducible, Cohen–Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of $X_{r,d,n}$ in terms of Castelnuovo–Mumford regularity and $d$-normality. This yields a characterization of the singular locus of $X_{2,d,n}$ and $X_{3,2,n}$.

Making Waves

We study linear PDE constraints for vector-valued functions and distributions. Our focus lies on wave solutions, which give rise to distributions with low-dimensional support. Special waves from vector potentials are represented by syzygies. We parametrize all waves by projective varieties derived from the support of the PDE. These include determinantal varieties and Fano varieties, and they generalize wave cones in analysis.

Borel–Moore homology of determinantal varieties

Borel–Moore homology of determinantal varieties

Tropical positivity and determinantal varieties

Tropical positivity and determinantal varieties

Linear codes associated to symmetric determinantal varieties: Even rank case

We consider linear codes over a finite field Fq, for odd q, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a codeword is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upperbounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order 5, are given.

Curvature loci of 3‐manifolds

AbstractWe refine the affine classification of real nets of quadrics in order to obtain generic curvature loci of regular 3‐manifolds in and singular corank one 3‐manifolds in . For this, we characterize the type of the curvature locus by the number and type of solutions of a system of equations given by four ternary cubics (which is a determinantal variety in some cases). We also study how singularities of the curvature locus of a regular 3‐manifold can go to infinity when the manifold is projected orthogonally in a tangent direction.

Standard monomials and invariant theory for arc spaces I: General linear group

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field [Formula: see text], we construct a standard monomial basis for the arc space of the determinantal variety over [Formula: see text]. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

Singularities of determinantal pure pairs

Let $X$ be a generic determinantal affine variety over a perfect field of characteristic $p \geq 0$ and $P \subset X$ be a standard prime divisor generator of $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ and so that $(X,P)$ is purely log terminal (PLT) if $p=0$ and $(X,P)$ is log $\mathbb{Q}$-Gorenstein. In general, using recent results of Z. Zhuang and S. Lyu, we show that $(X,P)$ is of PLT-type, i.e. there is a $\mathbb{Q}$-divisor $\Delta$ with coefficients in $[0,1)$ such that $(X,P+\Delta)$ is PLT.

Mixed Hodge Structure on Local Cohomology with Support in Determinantal Varieties

Abstract We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.