American Journal of Mathematics, Volume 128, 2006 - Table of Contents

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 C^{1,γ} [Access article in PDF] Abstract:

We prove C^{1},γ regularity of Lipschitz free boundaries of two-phase problems for a class of
homogeneous fully nonlinear elliptic operators F(D^{2}u(x), x) with Hölder dependence on x, containing
convex (concave) operators.
(pages 541-571.)
Abstract in Tex

Let S = K[x_{1}, . . . , x_{n}], let A, B be finitely generated graded S-modules, and let m =
(x_{1}, . . . , x_{n}) ⊂ S. We give bounds for the regularity of the local cohomology of Tor_{k} (A, B) in terms
of the graded Betti numbers of A and B, under the assumption that dim Tor_{1} (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 /2 steps of the
resolution of I are linear, then I^{2} 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 [Access article in PDF] Abstract:

Let SU_{C}(r) be the moduli space of vector bundles of rank r and trivial determinant on
a curve C. A general E in SU_{C}(r) defines a divisor Θ_{E} in the linear system |rΘ|, where Θ is
the canonical theta divisor in Pic^{g-1} (C). This defines a rational map θ: SU_{C}(r) → |rΘ|, which
turns out to be the map associated to the determinant bundle on SU_{C}(r) (the positive generator
of Pic (SU_{C}(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 SU_{C}(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 [Access article in PDF] 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 GL_{n}, 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 VII with numerically anticanonical divisor [Access article in PDF] Abstract:

We consider minimal compact complex surfaces S with Betti numbers b_{1} = 1 and n =
b_{2} > 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 H^{0}(S,-mK ⊗ F) ≠ 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

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

In a previous paper with Schmid we considered the regularity of automorphic distributions
for GL(2,), 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 ,
uniformly in α ∈ , for an the coefficients of the L-function of a cusp form on GL(3, )\GL(3,).
We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp
forms on GL(n,), 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 [Access article in PDF] Abstract:

The three-dimensional Schrödinger propogator e^{itH} , 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 L^{p} 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 [Access article in PDF] 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 .
(pages 751-786.)
Abstract in Tex

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