288 research outputs found
Some remarks on the work of Lawrence Ein
This is a write-up of introductory remarks that I made at the UIC conference
in honor of Lawrence Ein's 60th birthday. It presents an informal survey of
some of Ein's work, interspersed with stories and reminiscences
An informal introduction to multiplier ideals
Multiplier ideals, and the vanishing theorems they satisfy, have found many
applications in recent years. In the global setting they have been used to
study pluricanonical and other linear series on a projective variety. More
recently, they have led to the discovery of some surprising uniform results in
local algebra.
The present notes aim to provide a gentle introduction to the
algebraically-oriented local side of the theory. They follow closely a short
course on multiplier ideals given in September 2002 at the Introductory
Workshop of the program in commutative algebra at MSRI.Comment: 28 pages, 5 figures, minor corrections and improvements according to
editors suggestion
A Geometric Effective Nullstellensatz
We present in this paper a geometric theorem which clarifies and extends in
several directions work of Brownawell, Kollar and others on the effective
Nullstellensatz. To begin with, we work on an arbitrary smooth complex
projective variety X, with previous results corresponding to the case when X is
projective space. In this setting we prove a local effective Nullstellensatz
for ideal sheaves, and a corresponding global division theorem for adjoint-type
bundles. We also make explicit the connection with the intersection theory of
Fulton and MacPherson. Finally, constructions involving products of prime
ideals that appear in earlier work are replaced by geometrically more natural
conditions involving order of vanishing along subvarieties. The main technical
inputs are vanishing theorems, which are used to give a simple
algebro-geometric proof of a theorem of Skoda type, which may be of independent
interest.Comment: Introduction expanded, examples added, work of Sombra discusse
The gonality conjecture on syzygies of algebraic curves of large degree
We show that a small variant of the methods used by Voisin in her study of
canonical curves leads to a surprisingly quick proof of the gonality conjecture
of Green and the second author, asserting that one can read off the gonality of
a curve C from its resolution in the embedding defined by any one line bundle
of sufficiently large degree. More generally, we establish a necessary and
sufficient condition for the asymptotic vanishing of the weight one syzygies of
the module associated to an arbitrary line bundle on C
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