Khazhgali Kozhasov
We prove that semialgebraic sets of rectangular matrices of a fixed rank, of skew-symmetric matrices of a fixed rank and of real symmetric matrices whose eigenvalues have prescribed multiplicities are minimal submanifolds of the space of real matrices of a given size.
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For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical K-trivial) degenerations. Although weak K-moduli is not unique in general, they always contain a unique partial compactification (K-moduli). Our main theorem is the log minimality of their normalizations, under some conditions. Partially to confirm that known examples satisfy the conditions, we also include an appendix on the algebro-geometric reconstruction of Kul…
On Stringy E-functions and the Non-negativity Conjecture for Determinantal Varieties
本文计算了行列式簇的stringy E-函数，证明其与Grassmannian和仿射空间乘积的E-函数一致，并据此验证了stringy Hodge数的非负性猜想。
On the monodromy conjecture for determinantal varieties
该研究通过分析单值化zeta函数、运用推广A'Campo公式等方法，证明了行列式簇的单值化猜想与全纯性猜想，还证明了一般曲线Brill-Noether轨迹的单值化猜想。
On area-minimizing Pfaffian varieties
本文证明了除 Pfaffian 超曲面外，所有 Pfaffian 簇均为面积极小的，推广了 Kerckhove 和 Lawlor 关于行列式簇的结果。
This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney's $m_{d}$ multiplicity.
We show that the log canonical threshold of a generic determinantal variety and its generic link are the same.
We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions.
Seiberg-like duality for resolutions of determinantal varieties
该研究针对行列式簇的PAX与PAXY两种自然消解的零亏格Gromov-Witten理论，通过箭图突变关联对应箭图丛，证明二者生成函数满足聚类变量替换，还得到量子Thom-Porteous公式与箭图丛的类Seiberg对偶结论。
A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a $m\times n$-matrix. In this note, we consider $\mathcal{G}$-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade. In this note, we prove that matrices parametrized by generic homogeneous forms of degree $d$ define EIDS. It follows that $\mathcal{G}$-finite determinacy of…
We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum α_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal subvarieties. Our result implies the semicontinuity conjecture for minimal log discrepancies of such pairs. For these computations, we use the description of minimal log discrepancies via codimensions of cylinders in the space of jets; this necessitates the computations of an explicit generator for the canonical differential forms and the Na…
Linear projections from P^k to P^h appear in computer vision as models of images of dynamic or segmented scenes. Given multiple projections of the same scene, the identification of many enough correspondences between the images allows, in principle, to reconstruct the position of the projected objects. A critical locus for the reconstruction problem is a variety in P^k containing the set of points for which the reconstruction fails. Critical loci turn out to be determinantal varieties. In this paper we determine and classify all the smooth critical loci, showing that they are classical project…
Nash blowups of 2-generic determinantal varieties in positive characteristic
本文通过显式描述toric结构并证明Nash吹胀组合学无特征依赖，得出正特征域上2-generic行列式簇的Nash吹胀非奇异。
We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.
The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes and Chern-Mather classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes.
There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition…
This is the author's 2008 thesis from the University of Chicago. We generalize the notion of the Clifford index to an arbitrary very ample line bundle and show how it determines when a curve and its various secant varieties have determinantal equations. In particular we establish the scheme theoretic version of a conjecture of Eisenbud-Koh-Stillman. If $L_1$ and $L_2$ are line bundles of degree at least $2g+1+k$, then ${\rm Sec}^j(C)$ is determinantally presented for $j \leq k$ and for $j \leq k+1$ if $L_1 \neq L_2$. We also give a geometric characterization of the standard Clifford index in t…
We study the family of ideals defined by mixed size minors of two-sided ladders of indeterminates. We compute their Groebner bases with respect to a skew-diagonal monomial order, then we use them to compute the height of the ideals. We show that these ideals correspond to a family of irreducible projective varieties, that we call mixed ladder determinantal varieties. We show that these varieties are arithmetically Cohen-Macaulay. We characterize the arithmetically Gorenstein ones, among those that satisfy a technical condition. Our main result consists in proving that mixed ladder determinanta…
This article studies the scheme structure of the jet schemes of determinantal varieties. We show that in general, these jet schemes are not irreducible. In the case of the determinantal variety $X$ of $r \times s$ matrices of rank at most one, we give a formula for the dimension of each of the components of its jet schemes. As an application, we compute the log canonical threshold of the pair $(\mathbb{A}^{rs},X)$.
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ 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 $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. 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…
We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.
In this paper we prove Homological Projective Duality for crepant categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a n x m matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we discuss the relation between rationality an…
We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a Lê-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last se…
Let Y be the variety of (skew) symmetric nxn-matrices of rank less than or equal to r. In paper we construct a full faithful embedding between the derived category of a non-commutative resolution of Y, constructed earlier by the authors, and the derived category of the classical Springer resolution of Y.
In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. As an applica…
We study the blow-ups X of P3 along a proj. normal curve C. We look for very ample divisor classes on X of low degree, and we study the ideal of the embedding of X. Some result is generalized to higher dimensions.
The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaff…
In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of matrices of co-rank r-1. In this paper, we prove that conjecture for rectangular matrices and for symmetric matrices, as well as a variant for skew-symmetric matrices. To appear in International Mathematics Research Notices.
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.
Hilbert series of second order jets of determinantal varieties
本文针对行列式簇射流概型中希尔伯特函数难以计算的问题，成功给出了 $2\ imes n$ 矩阵极大子式情形下二阶射流概型的希尔伯特函数。
We record an observation about the Witten indices in two families of gauged linear sigma models: the U(2) model for linear sections of Grassmannians, and the U(1) model for quadric complete intersections. We describe how the Witten indices are related to the Euler characteristics of the singular skew-symmetric or symmetric determinantal varieties featuring in the analysis of their opposite phases, and we discuss the extent to which these relationships can be reconciled with standard Born-Oppenheimer arguments.
We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors to zero. These varieties can be interpreted as generalized tangent bundles over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when $r = m$, the varieties are irreducible, but when $r < m$, these varieties have at least $\lfloor {k/2}\rfloor + 1$ components. In fact, w…
Ulrich sheaves and determinantal representations for higher secant varieties of curves
研究光滑射影曲线高割线簇的行列式表示，利用高阶Szegő核等工具，在嵌入充裕时给出对称表示，对实曲线割线簇给出正定条件，对有理正规曲线割线簇给出明确表示，还能在特定条件下将曲线凸包表示为谱面体。
Gröbner bases for bipartite determinantal ideals
该研究提出初等证明方法，借助基于行列式莱布尼茨公式的S-多项式构造，证明二分行列式理想的自然生成元构成Gröbner基，并探讨其在张量与代数统计中的应用。
A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal d…
Decomposing Determinantal Varieties from Statistics via Matroid Theory
本文通过模体理论研究行列式变种的分解，提供系统描述和计算次数的组合框架，扩展了当前计算技术的范围。
Real and Positive Tropicalizations of Symmetric Determinantal Varieties
本文研究了Puiseux级数上低秩对称矩阵簇的实与正热带化，证明了实与复热带化在秩二和余秩一情形下一致，并揭示了Speyer和Williams提出的两种正热带化概念在相同条件下的异同。
We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights.…
We examine a class of Calabi-Yau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus-0 Gromov-Witten invariants from the analysis of the Givental $I$-functions. By constructing $I$-functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus-0 A-model correlation functions appropriately. We also check that our results for the Gromov-Witten invariants are consistent with previous results for …
The Euler characteristic of Milnor fibers over 2-generic symmetric determinantal varieties
本文推导了非退化函数f:X→C的米尔诺纤维欧拉特征公式，通过描述概形的托尔结构和计算牛顿多面体体积，应用Matsui-Takeuchi公式得出结果，并探讨了局部欧拉阻挠与多项式米尔诺数的关系。
Ladder determinantal varieties and their symbolic blowups
本文研究了混合阶梯行列式理想符号Rees代数的F-正则性，证明其关联格拉德代数的F-纯性，并展示了minors poset理想产生F-纯代数的性质。
Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus
本文提出了一种张量展开方法，用于研究secant种群的行列式方程，并证明了2×2矩阵乘法张量的边界秩为7。
This paper proves that when the $r \times r$ minors of an $m \times n$ matrix of indeterminates are not a tropical basis then the tropical prevariety has greater dimension than the tropical variety. It proves the same for the $r \times r$ minors of an $n \times n$ symmetric matrix of indeterminates when r > 4.
On jet schemes of determinantal varieties
本文利用jet scheme方法研究行列式簇，发现了其与jet schemes间Hilbert级数的新联系，证明了特定情形下定义理想的可壳性，并为一般情形提供了定义不可约分支的多项式族。
We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.
We introduce certain quiver analogue of the determinantal variety. We study the Kempf-Lascoux-Weyman's complex associated to a line bundle on the variety. In the case of generalized Kronecker quivers, we give a sufficient condition on when the complex resolves a maximal Cohen-Macaulay module supported on the quiver determinantal variety. This allows us to find the set-theoretical defining equations of these varieties. When the variety has codimension one, the only irreducible polynomial function is a relative tensor invariant. As a by-product, we find some vanishing condition for the Kronecker…
We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the Čech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology …
The two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi--Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models -- which we refer to as the PAX and the …
On stationary real matrix Schubert varieties
本文研究了实矩阵施布特簇的平稳性，证明了特定条件下其极小性，并揭示了相关簇的结构特性。
Singularities of orthogonal and symplectic determinantal varieties
本文研究正交与辛行列式簇的奇点，证明辛情形下轨道闭包正规且具有有理奇点，并精确刻画正交情形下的例外情况。
We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to p…
This work is entirely devoted to construct huge families of indecomposable arithmetically Cohen-Macaulay (resp. Ulrich) sheaves E of arbitrary high rank on a general standard (resp. linear) determinantal scheme X\subset \PP^n of codimension c \ge 1, n-c \ge 1 and defined by the maximal minors of a t \times (t+c-1) homogeneous matrix A. The sheaves E are constructed as iterated extensions of sheaves of lower rank. As applications: (1) we prove that any general standard determinantal scheme X\subset \PP^n is of wild representation type provided the degrees of the entries of the matrix A satisfy …
For $m\geq n$, Let $K$ be an algebraic closed base field, and define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called determinantal variety. In most cases $τ_{m,n,k}$ is singular with singular locus $τ_{m,n,k+1}$. In this paper we compute the local Euler obstruction of $τ_{m,n,k}$, and we prove that the characteristic cycle of the intersection cohomology complex of $τ_{m,n,k}$ is irreducible. We also give an explicit formula for the Chern-Mather class of $τ_{m,n,k}$ as a clas…
Let X be a standard determinantal scheme X \subset \PP^n of codimension c, i.e. a scheme defined by the maximal minors of a t \times (t+c-1) homogeneous polynomial matrix A. In this paper, we study the main features of its normal sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a line bundle \shL on X \setminus Sing(X) such that \shN_X \otimes \shL is arithmetically Cohen-Macaulay and, even more, it is Ulrich whenever the entries of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and, finally, (3) \shN_X is μ-(semi)stable provided the entries of A are…
Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case
本文研究奇特征有限域上由有界秩对称矩阵行列式簇导出的线性码，推导码重公式并计算偶数秩上界情形的最小距离，提出奇数情形猜想，并给出权重表及勘误。
Determinantal ideals of secant varieties
本研究利用点Hilbert概型，证明了极充足嵌入下光滑射影簇的割线簇理想的行列式表示性质，解决了多个相关猜想，并验证了三点Hilbert概型的爆破结构。
Determinantal varieties from point configurations on hypersurfaces
本文研究参数化共处于同一 $d$ 次超曲面上的 $n$ 个有序射影点的概形 $X_{r,d,n}$，揭示其行列式结构并证明不可约、Cohen–Macaulay 与正规性，还刻画了其奇异轨迹。
We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2 x 2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number …
Let $K$ be an algebraically closed field of characteristic $0$. For $m\geq n$, we define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\bbP^{mn-1}$, and is called the (generic) determinantal variety. In most cases $τ_{m,n,k}$ is singular with singular locus $τ_{m,n,k+1}$. In this paper we give explicit formulas computing the Chern-Mather class ($c_M$) and the Chern-Schwartz-MacPherson class ($c_{SM}$) of $τ_{m,n,k}$, as classes in the projective space. We also obtain formulas for the conormal cycles and the charac…
Linear codes associated to symmetric determinantal varieties; General case
本文研究奇数有限域上由有界秩对称矩阵定义的线性码，建立码字权重与对称矩阵结合方案 Q-数的联系，得到权重分布公式，并统一确定奇偶秩界下的最小距离。
The vector space of m x n complex matrices (m >= n) admits a natural action of the group GL = GL_m x GL_n via row and column operations. For positive integers a,b, we consider the ideal I_{a x b} defined as the smallest GL-equivariant ideal containing the b-th powers of the a x a minors of the generic m x n matrix. We compute the syzygies of the ideals I_{a x b} for all a,b, together with their GL-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors (b=1), and of Akin-Buchsbaum-Weyman for the powers of the ideals of maximal minors (a=n). Our methods rely on a…
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