Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices

Determinantal representations and the image of the principal minor map

In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $({\rm SL}_2(\mathbb{R}))^{n} \rti…

实对称矩阵主子式的符号模式

Sign patterns of principal minors of real symmetric matrices

本文分析了实对称矩阵主子式符号满足的组合规则，证明其等价于一致定向 Lagrangian 拟阵，揭示了其结构、对称性及渐近不可表示性，并提出了关于可表示符号模式及其表示空间拓扑的猜想与实验结果。

主子式映射的纤维

The Fiber of the Principal Minor Map

本文研究了主子式映射在一般域上的纤维，给出了纤维在对角等价意义下为单点的充要条件，并利用矩阵可约性与行列式表示的联系刻画了对称与Hermitian矩阵的纤维，同时回答了关于实稳定矩阵的遗留问题。

On binomial equations defining rational normal scrolls

We show that a rational normal scroll can in general be set-theoretically defined by a proper subset of the 2-minors of the associated two-row matrix. This allows us to find a class of rational normal scrolls that are almost set-theoretic complete intersections.

Segre 簇的切触簇的集合论定义方程

Set-theoretic defining equations of the tangential variety of the Segre variety

本文证明了 Landsberg--Weyman 猜想的集合论版本，通过引入排他秩概念，将 Segre 簇的切触簇表示为排他秩不超过一的对称矩阵的主子式簇，从而给出其定义方程。

图态与主子式簇

Graph States and the Variety of Principal Minors

本文揭示了图态与二元对称主子式簇之间的对应关系，通过将局部Clifford群的作用转化为局部辛群在Lagrangian Grassmannian上的作用，建立了量子稳定子态与代数几何簇的深刻联系。

Box-shaped matrices and the defining ideal of certain blowup surfaces

We study the defining equations of projective embeddings of the blowup of P^2 at a set of {d+1 \choose 2} number of points in generic position. To do this, we first generalize the notion of a matrix, its ideal of 2x2 minors to that of a box-shaped matrix. Our work completes previous works of Geramita and Gimigliano.

On eigenvalues of symmetric matrices with PSD principal submatrices

We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ is fixed) minors are positive semidefinite. It is proven that the set $λ(\mathcal{S}^{n,k})$ of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant $\mathbb{R}^n_{\geq 0}$ and not convex already when $(n,k)=(4,2)$.

Minimal free resolutions of ideals of minors associated to pairs of matrices

Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$ and $\operatorname{rank} (A\cdot B) \leq c$, for three non-negative integers $a,b,c$. These varieties are precisely the orbit closures of representations for the equioriented $A_3$ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of…

Determinantal ideals and monomial curves in the three-dimensional space

We show that the defining ideal of every monomial curve in the affine or projective three-dimensional space can be set-theoretically defined by three binomial equations, two of which set-theoretically define a determinantal ideal generated by the 2-minors of a $2\times 3$ matrix with monomial entries.

Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

Hilbert series of certain jet schemes of determinantal varieties

We consider the affine variety ${\mathcal{Z}_{2,2}^{m,n}}$ (or just "$Y$") of first order jets over ${\mathcal{Z}_{2}^{m,n}}$ (or just "$X$"), where $X$ is the classical determinantal variety given by the vanishing of all $2\times 2$ minors of a generic $m\times n$ matrix. When $2 < m \le n$, this jet scheme $Y$ has two irreducible components: a trivial component, isomorphic to an affine space, and a nontrivial component that is the closure of the jets supported over the smooth locus of $X$. This second component is referred to as the principal component of $Y$; it is, in fact, a cone and can …

Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain cohomology modules vanish, a classical problem raised by Grothendieck. In the second chapter we will use local cohomology to study the connectedness behavior during a Groebner deformation and the arithmetical rank of certain varieties. In the third chapter, we will investigate the relations between the minors of a fixed size of a generic matrix by using too…