Computation of the aa-invariant of ladder determinantal rings

We solve the problem of effectively computing the $a$-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.Comment: AmS-LaTeX, 20 pages; minor improvements of presentatio

Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gr\"obner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo-Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind-Mertens formula for the content of the product of two polynomials.Comment: 36 pages, 9 figures. In the updated version, section 7: "Final remarks and open problems" is new; the introduction was updated accordingly. References update

Posets of h-vectors of standard determinantal schemes

We study the combinatorial structure of the poset H consisting of h-vectors of length s of codimension c standard determinantal schemes, defined by the maximal minors of a t × (t + c − 1) homogeneous, polynomial matrix. We show that H obtains a natural stratification, where each strata contains a maximum h-vector. Moreover, we prove that any h-vector in H is bounded from above by a h-vector of the same length and which corresponds to a codimension c level standard determinantal scheme. Furthermore, we show that the only strata in which there exists also a minimum h-vector is the one consisting of h-vectors of level standard determinantal schemes

Schubert Numbers

This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders

Hilbert Functions of Ladder Determinantal Varieties

We consider algebraic varieties de#ned by thvanish,f ofall minors ofa #xed size ofa rectangular matrixwith indeterminate entriessuch thh th indeterminates inth9K minors are restricted to lie in a ladder shder region of th rectangular array. Explicit formulae for th Hilbert function ofsuch varieties are obtained in (i) th rectangular case byAbhKT`Lf (Rend. Sem. Mat. Univers. Politecn. Torino 42 (1984) 65), and (ii)th case of22 minors in one-sided ladders by Kulkarni (Semigroup ofordinary multiple point, analysis ofstraighL8`fE formula and counting monomials,Phno Thnomi Purdue University, West Lafayette, USA, 1985). More recently,KrattenthKUK andProhL9` (Trans. Amer. Math Soc. 351 (1999) 1015)h15 proved a `remarkable formula', conjectured by Conca and Herzog (Adv. Math 132 (1997) 120) for th Hilbert series in th case ofarbitrary sized minors in one-sided ladders. We describehsc an explicit, albeit complicated, formula for th Hilbert function and th Hilbert series in th case of arbitrary sized minors in two-sided ladders. From a combinatorial viewpoint,the is equivalent to th enumeration ofcertain sets of`indexed monomials'. c 2002 Elsevier Science B.V. All righ8 reserved. MSC: Primary 05A15; 13C40; 13D40; 14M12; Secondary 05A19; 05E10; 14M15 Keywords: Hilbert functions; Hilbert series; Determinantal varieties; Ladder determinantal ideals; Indexed monomials 0. I9363Dg1404 Let K be a #eld and X =(X ij )beanm(1)m(2) matrixwhri entries are variables over K . Given a subset Yofth integral rectangle [1;m(2)] = {(i; j) :16 i 6m(1) and 1 6 j 6m(2)}; # A partofthP work was supported byresearch grant No. 93-106=RG=MATHS=AS fromth Thmf World Academy ofSciences, Italy. Currently,th authe is partially supported by a `Career Award' grant from AICTE, NewDelh and an IRCC gran..