The hilbert function and the minimal free resolution of some gorenstein ideals of codimension 4

We give a construction for Gorenstein ideals of codimension 4 which we believe have maximal graded Betti numbers for their Hilbert function.     

Finite p-Groups All of Whose Proper Quotient Groups are Abelian or Inner-Abelian

In this article, finite p-groups all of whose proper quotient groups are abelian or inner-abelian are classified. As a corollary, finite p-group all of whose proper quotient groups are abelian, and finite p-groups all of whose proper sections are abelian or inner-abelian are also classified.     

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).     

Quasi-Piecewise-Koszul Modules

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

Some results about zero‐cycles on abelian and semi‐abelian varieties

In this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.     

Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties

Abstract

Following Contou-Carrère (Contou-Carrère,C. (1983). Géométrie des Groupes Semi-Simples,Résolutions équivariantes et Lieu Singulier de Leurs Variétés de Schubert. Thèse d’état,Université Montpellier II (published partly as,Le Lieu singulier des variétés de Schubert (1988). Adv. Math.,71:186–221)),we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. Using Carrell and Peterson's characterization (Carrell,J. B. (1994). The Bruhat graph of a Coxeter group,a conjecture of Deodhar,and rational smoothness of Schubert varieties. Proc. Symp. in Pure Math. 56(Part I):53–61),we prove that rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called,the combinatorial tangent space.