Gale duality and Koszul duality

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.     

On rationalizing divisors

Rational pairs generalize the notion of rational singularities to reduced pairs (X, D). In this paper we deal with the problem of determining whether a normal variety X has a rationalizing divisor, i.e., a reduced divisor D such that (X, D) is a rational pair. We give a criterion for cones to have a rationalizing divisor, and relate the existence of such a divisor to the locus of rational singularities of a variety.     

Wolfe duality and Mond-Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond-Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.     

Quasi-free quantum stochastic integrals for the CAR and CCR

A theory of quantum martingales and quantum stochastic integrals in quasi-free representations of the CAR and CCR is presented. For the CAR, the results generalize some of those developed in Barnett, Streater, and Wilde (J. Funct. Anal.48 (1982), 172–212, J. London Math. Soc.27 (1983), 373–384) and for the CCR, the results contain the standard Itô theory of stochastic integration with respect to Brownian motion as a special case.     

Inner divisors and composition operators

This characterization is stated and proved in much greater generality, beginning with subspaces of arbitrary L^∞ spaces and using a notion of inner divisors. Among the consequences are a variant of Wermer's theorem on embedding disks in maximal ideal spaces and a result on linear isometrics of H^∞. The ranges of composition operators C_I when I is not necessarily inner are characterized. In particular, relatively closed sets E ?cΔ of zero logarithmic and zero analytic capacity are characterized in terms of the algebras of bounded analytic functions on A invariant under corresponding Fuchsian groups. The paper concludes with an example of a uniformly closed subalgebra of H^∞ which contains the constants and the inner factors of its members, but is not of the form H ∞ I.     

Dicritical divisors and Jacobian problem

The dicritical divisor coming out of integrable holonomic systems gets miraculously married to the jacobian problem of algebraic geometry.     

Ample subvarieties and q-ample divisors

We introduce a notion of ampleness for subschemes of any codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti–Grauert vanishing theorem.     

Binz-Butzmann duality versus Pontryagin duality

Supported by Ministerio de Education y Ciencia, grant #BE91-031     

Equivalence without determinantal divisors

A simple number-theoretic lemma is proved, and then used to develop the theory of equivalence for matrices over principal ideal rings without recourse to the concept of determinantal divisors.     

On partially ample divisors

Partially ample divisors are defined by relaxing the different conditions that characterize the ample divisors. We prove that for nef and big divisors such notions coincide. We also prove that the partial ampleness of big divisors are preserved in the positive parts of the Fujita‐Zariski decomposition.     

Prime divisors of palindromes

Summary In this paper, we study some divisibility properties of palindromic numbers in a fixed base <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>g\ge 2$. In particular, if ${\mathcal P}_L$ denotes the set of palindromes with precisely $L$ digits, we show that for any sufficiently large value of $L$ there exists a palindrome $n\in{\mathcal P}_L$ with at least $(\log\log n)^{1+o(1)}$ distinct prime divisors, and there exists a palindrome $n\in{\mathcal P}_L$ with a prime factor of size at least $(\log n)^{2+o(1)}$.     

Limits of Cartier divisors

Consider a one-parameter family of algebraic varieties degenerating to a reducible one. Our main result is a formula for the fundamental cycle of the limit subscheme of any family of effective Cartier divisors. The formula expresses this cycle as a sum of Cartier divisors, not necessarily effective, of the components of the limit variety.     

Cartier divisors and geometry of normalG-varieties

We study Cartier divisors on normal varieties with the action of a reductive groupG. We give criteria for a divisor to be Cartier, globally generated and ample, and apply them to a study of the local structure and the intersection theory of aG-variety. In particular, we prove an integral formula for the degree of an ample divisor on a variety of complexity 1, and apply this formula to computing the degree of a closed 3-dimensional orbit in any SL₂-module.Supported by CRDF grant RM1-206 and INTAS grant INTAS-OPEN-97-1570     

Homology of homogeneous divisors

This work deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors not coming from either hyperplane arrangements or discriminants in singularity theory.