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.     

On Characteristic Classes of Determinantal Cremona Transformations

We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes of rational polytopes. Dedicated to the memory of Shiing-Shen Chern     

Arithmetically Buchsbaum divisors on varieties of minimal degree

In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.

     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?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×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets 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 are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P⁶³ where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.     

Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

The aim of this paper is to study mixed multiplier ideals associated with a tuple of ideals in a two‐dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.     

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals I_t of the extended Hankel matrices. Denote by c-chain a sequence a₁,…,a_k with a_i+c<a_i+1 for all i=1,…,k−1. Using the results of c-chain, we solve the membership problem for the symbolic powers and we compute the primary decomposition of the product I_t1?I_tk of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product I_t1?I_tk has a linear resolution and the multi-homogeneous Rees algebra is defined by a Gröbner basis of quadrics.     

Markov processes on partitions

We introduce and study a family of Markov processes on partitions. The processes preserve the so-called z-measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal. Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time– dependent models.     

Laplace equations,Lefschetz properties and line arrangements

In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6]. The starting point was an example in [3]. Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3].     

Initial complex associated to a jet scheme of a determinantal variety

We show in this paper that the principal component of the first-order jet scheme over the classical determinantal variety of m×n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.     

The recursive nature of cominuscule Schubert calculus

The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.     

Betti numbers of mixed product ideals

We compute the Betti numbers of the resolution of a special class of square-free monomial ideals, the ideals of mixed products. Moreover when these ideals are Cohen-Macaulay we calculate their type. Received: 9 March 2008     

Algebraic properties of classes of path ideals of posets

We consider path ideals associated to special classes of posets such as tree posets and cycles. We express their property of being sequentially Cohen–Macaulay in terms of the underlying poset. Moreover, monomial ideals, which arise in algebraic statistics from the Luce-decomposable model and the ascending model, can be viewed as path ideals of certain posets. We study invariants of these so-called Luce-decomposable monomial ideals and ascending ideals for diamond posets and products of chains. In particular, for these classes of posets, we explicitly compute their Krull dimension, their projective dimension, their Castelnuovo–Mumford regularity and their Betti numbers.     

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

Mathematische Zeitschrift - 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...