On linear spaces of skew-symmetric matrices of constant rank
Laurent Manivel, Emilia Mezzetti
我们描述了六阶常秩四的复反对称矩阵的射影平面空间。我们证明了它有四个连通分支，在PGL_6的作用下都是26维的齐次连通分支。
We describe the space of projective planes of complex skew-symmetric matrices of order six and constant rank four. We prove that it has four connected components, all of dimension 26 and homogeneous under the action of PGL_6.
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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.
Vector spaces of skew-symmetric matrices of constant rank
本文研究了特征为零的代数闭域上常秩2 r（N+1）x（N +1）型斜对称矩阵向量空间在SL（N+1）自然作用下的轨道.我们给出了2维向量空间的轨道的完整描述，并将它们与某些线性型的1-一般矩阵联系起来。我们还证明了，对于P^2上的每个秩为2的向量丛，定义了P^2在G（1，7）中的三重Veronese嵌入，存在一个由8 × 8的常秩6的反对称矩阵组成的向量空间，其核丛是给定秩为2的向量丛的对偶.
Orthogonal bundles and skew-Hamiltonian matrices
利用斜Hamilton矩阵的性质和经典的连通性结果，证明了$\mathbb{P}^2$上的稳定秩$r$正交向量丛的模空间$M_{ort}^0(r,n)$，其Chern类为$(c_1,c_2)=(0,n)$，平凡分裂在一般直线上，对于$r$和$n$的特定值，是维数$(r-2)n-{r \choose 2}$的光滑不可约空间.
It is well known that every smooth cubic threefold is the zero locus of the Pfaffian of a 6 x 6 skew-symmetric matrix of linear forms in P^4. To compactify the space of such Pfaffian representations of a given cubic and to study the construction in families as well as for singular or reducible cubics, it is thus natural to consider the incidence correspondence of Pfaffian representations inside the product of the space of semistable skew-symmetric 6 x 6 matrices of linear forms in P^4 and the space of cubics. Here we describe concretely the irreducible component of this incidence correspondenc…
We show that every skew-symmetric 6 x 6 matrix of linear forms with vanishing Pfaffian is congruent to one of finitely many types of matrices, each of which is characterised by a specific pattern of zeroes (and some other linear relations) among its entries. Such matrices are for example important for compactifying moduli spaces of stable rank 2 vector bundles with Chern classes c_1=0, c_2=2 on cubic threefolds.
The Decomposition Algorithm of Skew-symmetrizable Exchange Matrices
一些斜对称整数交换矩阵与标记有边曲面的理想（标记）三角剖分相关联。这些交换矩阵允许反对称矩阵的展开。我们开发了一个组合算法，确定如果一个给定的斜对称矩阵是这样的类型。该算法推广了\cite{WG}中的算法。作为推论，我们用这个算法来确定一个给定的可斜对称化矩阵是否具有有限突变类型。
Skew-symmetric matrices and Palatini scrolls
证明了当m大于3，k大于m-2时，2k维向量空间上的斜对称形式空间的m维子空间的格拉斯曼算子对于P^（2k-1）中Palatini涡卷的Hilbert格式是双有理的.当m=3，k大于3时，证明了这个Grassmannian对于（E，Y）的集合是双有理的，其中Y是k次光滑平面曲线，E是Y上行列式为O（k-1）的稳定秩2丛.
Border rank=rank for Kruskal tensors and a Kruskal's theorem for skew decompositions
该研究证明Kruskal张量的边界秩等于秩，给出交替张量的Kruskal条件以证明斜秩分解唯一性，还证明交替Kruskal张量的边界斜秩等于斜秩并给出最小斜秩分解算法。
Given a 6-dimensional complex vector space $W$, we consider linear systems of skew-symmetric forms on W. The $n$-dimensional linear systems this kind, that can also be interpreted as $n$-dimensional linear subspaces of $\mathbb{P}(\bigwedge^2 W^*)$, are parametrized by the projective space $\mathbb{P}(\mathbb{C}^{n+1}\otimes \bigwedge ^2 W^*)$. We analyze the $SL(W)$ action on this projective space and the GIT stability of linear systems with respect to this action. We present a classification of all stable orbits of linear systems whose generic element is a tensor of rank 4.
Cluster-cyclic condition of skew-symmetrizable matrices of rank 3 via the Markov constant
本文针对3阶斜对称可化矩阵，结合马尔可夫常数研究其对应加权箭图经突变后始终保持循环的条件，拓展了马尔可夫常数在斜对称矩阵相关研究中的应用场景。
The main result of the paper is a classification of singularities of complex skew-symmetric matrix families of even size which are simple under a natural equivalence relation. The classification is obtained by appropriate suspensions of simple families of arbitrary square matrices classified by Bruce and Tari. The suspensions we are using are based on the embedding of a space of arbitrary square matrices into the space of skew-symmetric matrices of twice the size. We also show that similar relations embed Bruce's classification of simple symmetric matrix families into the Bruce-Tari simple lis…
We introduce the ``skew apolarity lemma'' and we use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in {$\bigwedge^dV_{\mathbb{C}}$ with $d\leq 3$ and $\dim V_{\mathbb{C}} \leq 8$}. New algorithms to compute the rank and a minimal decomposition of a tritensor are also presented.
Nonemptiness of skew-symmetric degeneracy loci
设V是d维复射影概形X上的秩N向量丛;设V具有反称双线性型，其值在线丛L中，且Λ^2 V^* \otimes L是充足的。假设形式在X的任意点的最大秩是r，其中r>0是偶数。本文的主要结果是：若d>2（N-r），则其秩至多为r-2的点的轨迹非空。这是数学AG/0305159的主要结果的斜对称模拟;证明是类似的。如果放宽了对满度的假设，我们得到了X的最大维数的一个较弱的估计（并给出了对称情况下的类似结果）。我们给应用程序的反对称矩阵的子计划，并分层的对偶的李代数的轨道维数。
Totally positive skew-symmetric matrices
本文重新定义了斜对称矩阵的完全正性，基于正交Grassmannian提供了正性判据，揭示了Pfaffian的符号模式，并引入了利用关联matroid确定胞腔归属的方法。
Moduli spaces of complete skew-forms are compactifications of spaces of skew-symmetric linear maps of maximal rank on a fixed vector space, where the added boundary divisor is simple normal crossing. In this paper we compute their effective, nef and movable cones, the generators of their Cox rings, and for those spaces having Picard rank two we give an explicit presentation of the Cox ring. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of some spaces of complete skew-forms having Picard rank at most four.
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