Symmetric ladders and G-biliaison

G-biliaison of ladder Pfaffian varieties

The ideals generated by pfaffians of mixed size contained in a subladder of a skew-symmetric matrix of indeterminates define arithmetically Cohen-Macaulay, projectively normal, reduced and irreducible projective varieties. We show that these varieties belong to the G-biliaison class of a complete intersection. In particular, they are glicci.

The G-biliaison class of symmetric determinantal schemes

We consider a family of schemes, that are defined by minors of a homogeneous symmetric matrix with polynomial entries. We assume that they have maximal possible codimension, given the size of the matrix and of the minors that define them. We show that these schemes are G-bilinked to a linear variety of the same dimension. In particular, they can be obtained from a linear variety by a finite sequence of ascending G-biliaisons on some determinantal schemes. In particular, it follows that these schemes are glicci. We describe the biliaisons explicitely in the proof of the main theorem.

Mixed ladder determinantal varieties from two-sided ladders

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…

Rigid ideals by deforming quadratic letterplace ideals

We compute the deformation space of quadratic letterplace ideals $L(2,P)$ of finite posets $P$ when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal $J(2,P)$ defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases $J(2,P)$ is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of cert…

Gröbner基、对称矩阵与C型Kazhdan-Lusztig簇

Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

研究一般对称矩阵子式生成的多项式理想，作为C型Kazhdan-Lusztig簇的概型定义理想，给出了其Gröbner基、初始理想素分解及基于pipe dream的Hilbert级数公式。