American Journal of Mathematics

American Journal of Mathematics
Volume 128, Number 3, June 2006

CONTENTS

    Ferrari, Fausto.
  • Two-phase problems for a class of fully nonlinear elliptic operators: Lipschitz free boundaries are C1,γ
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    Abstract:
      We prove C1,γ regularity of Lipschitz free boundaries of two-phase problems for a class of homogeneous fully nonlinear elliptic operators F(D2u(x), x) with Hölder dependence on x, containing convex (concave) operators. (pages 541-571.) Abstract in Tex
    Eisenbud, David.
    Huneke, Craig
    Ulrich, Bernd.
  • The regularity of Tor and graded Betti Numbers
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    Abstract:
      Let S = K[x1, . . . , xn], let A, B be finitely generated graded S-modules, and let m = (x1, . . . , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first 01/2 steps of the resolution of I are linear, then I2 is a power of m. (pages 573-605.) Abstract in Tex
    Beauville, Arnaud.
  • Vector bundles and theta functions on curves of genus 2 and 3
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    Abstract:
      Let SUC(r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C. A general E in SUC(r) defines a divisor ΘE in the linear system |rΘ|, where Θ is the canonical theta divisor in Picg-1 (C). This defines a rational map θ: SUC(r) → |rΘ|, which turns out to be the map associated to the determinant bundle on SUC(r) (the positive generator of Pic (SUC(r)). In genus 2 we prove that this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta map is a finite morphism - in other words, every vector bundle in SUC(3) admits a theta divisor. (pages 607-618.) Abstract in Tex
    Gelbart, Stephen S.
    Lapid, Erez M.
  • Lower bounds for L-functions at the edge of the critical strip
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    Abstract:
      We prove a coarse lower bound for L-functions of Langlands-Shahidi type of generic cuspidal automorphic representations on the line Re (s) = 1. We follow the path suggested by Sarnak using Eisenstein series and the Maass-Selberg relations. The bounds are weaker than what the method of de la Vallée Poussin gives for the standard L-functions of GLn, but are applicable to more general automorphic L-functions. Our Theorem answers in a strong form a conjecture posed by Gelbart and Shahidi [J. Amer. Math. Soc. 14 (2001)], and sharpens and considerably simplifies the proof of the main result of that paper (pages 619-638.) Abstract in Tex
    Dloussky, Georges.
  • On surfaces of class VII01 with numerically anticanonical divisor
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    Abstract:
      We consider minimal compact complex surfaces S with Betti numbers b1 = 1 and n = b2 > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m ≥ 1 and a flat line bundle F such that H0(S,-mKF) ≠ 0, then S contains a Global Spherical Shell. We apply this last result to complete classification of bihermitian surfaces. (pages 639-670.) Abstract in Tex
    Catanese, Fabrizio
    Hosten, Serkan.
    Khetan, Amit.
    Sturmfels, Bernd.
  • The maximum likelihood degree
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    Abstract:
      Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients. (pages 671-697.) Abstract in Tex
    Miller, Stephen D.
  • Cancellation in additively twisted sums on GL(n)
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    Abstract:
      In a previous paper with Schmid we considered the regularity of automorphic distributions for GL(2,r), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound equation, uniformly in α ∈ r, for an the coefficients of the L-function of a cusp form on GL(3,r )\GL(3,z). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,r), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions. (pages 699-729.) Abstract in Tex
    Goldberg, M.
  • Dispersive estimates for the three-dimensional Schrödinger equation with rough potentials
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    Abstract:
      The three-dimensional Schrödinger propogator eitH , H = - △ + V, is a bounded map from L1 to L with norm controlled by |t|-3/2 provided the potential satisfies two conditions: An integrability condition limiting the singularities and decay of V, and a zero-energy spectral condition on H. This is shown by expressing the spectral measure of H in terms of its resolvents and proving a family of Lp mapping estimates for the resolvents. Previous results in this direction had required V to satisfy explicit pointwise bounds. (pages 731-750.) Abstract in Tex
    Coskun, Izzet.
  • The enumerative geometry of Del Pezzo Surfaces via degenerations
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    Abstract:
      This paper investigates low-codimension degenerations of Del Pezzo surfaces. As an application we determine certain characteristic numbers of Del Pezzo surfaces. Finally, we analyze the relation between the enumerative geometry of Del Pezzo surfaces and the Gromov-Witten invariants of the Hilbert scheme of conics in r. (pages 751-786.) Abstract in Tex
    Mihalcea, Constantin.
  • Positivity in equivariant quantum Schubert calculus
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    Abstract:
      A conjecture of D. Peterson, proved by W. Graham, states that the structure constants of the (T-)equivariant cohomology of a homogeneous space G/P satisfy a certain positivity property. In this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology. (pages 787-803.) Abstract in Tex



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