Papers by Robert Lazarsfeld
Annales Scientifiques De L Ecole Normale Superieure, May 29, 2008
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associa... more In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.
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Eprint Arxiv Math 0506557, Jun 1, 2005
Let $\mathfrak a$ be an ideal of holomorphic functions vanishing only at the origin in $\mathbb{C... more Let $\mathfrak a$ be an ideal of holomorphic functions vanishing only at the origin in $\mathbb{C}^n$. The \textit{type} of $\mathfrak a$ is an invariant that measures the order of vanishing of the functions in $\mathfrak a$ along holomorphic curves; this invariant is of importance in the study of subelliptic estimates and subelliptic multiplier ideal sheaves. Recently there has been some interest in the question of which curves actually compute the type. In this note we prove that it is computed by one of the analytic irreducible components of the intersection of $n-1$ general functions in $\mathfrak a$.
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Inventiones Mathematicae, Aug 10, 2010
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Compositio Mathematica, 1988
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Invent Math, 2010
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The past decade has witnessed two important new developments in the study of linear series on alg... more The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on curves and surfaces. More recently, the viewpoints and techniques of higher dimensional geometry have started to play a central role. The purpose of these lectures is to provide a down-to-earth introduction to some of these new ideas and methods. The present notes, written with the assistance of G. Fernandez del Busto, are a considerably expanded version of a course given at the 1993 Regional Geometry Institute on Higher Dimensional Complex Geometry at Park City, Utah. While I hope they may be of interest to any geometers wishing to learn about recent work on linear series, I have particularly attempted to aim the discusion at a novice audience. My hope is that with a little faith and effort, these notes should be accessible to anyone having finished the standard introductory texts in algebraic geometry. The underlying theme of the lectures is the search for higher dimensional generalizations of the most basic facts about linear series on curves. But in an attempt to keep things elementary we work more or less exclusively on surfaces. Thus we end up discussing one central result -- viz. Reider's theorem -- from many points of view. Sticking to the case of surfaces allows one to eliminate many technical complexities, and I hope that parts of these lectures might therefore serve as a useful first introduction to the powerful cohomological techniques (involving $Q$-divisors) of contemporary
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These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an i... more These notes are the write-up of my 2008 PCMI lectures on multiplier ideals. They aim to give an introduction to the algebro-geometric side of the theory, with an emphasis on its global aspects. The focus is on concrete examples and applications. The lectures take into account a number of recent perspectives, including adjoint ideals and the resulting simplifications in Siu's theorem on plurigenera in the general type case. While the notes refer to my book [PAG] and other sources for some technical points, the conscientious reader should arrive at a reasonable grasp of the machinery after working through these lectures.
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Lecture Notes in Mathematics, 1991
Without Abstract
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Lectures on Riemann Surfaces, 1989
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Publications mathématiques de l'IHÉS, 2015
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Lecture Notes in Mathematics, 1984
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Lecture Notes in Mathematics, 1984
Without Abstract
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Lecture Notes in Mathematics, 1981
... for local complete intersections. Taking Y = X c ~m , one finds that ~i(P m , X) = 0 for i &a... more ... for local complete intersections. Taking Y = X c ~m , one finds that ~i(P m , X) = 0 for i < 2n - m + 1 , which strengthens results of Barth, Larsen, and Ogus [5, 7, 44, 52]. Here is an overview of the contents and organization of these notes. ...
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Enumerative Geometry and Classical Algebraic Geometry, 1982
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Trends in Commutative Algebra, 2004
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Papers by Robert Lazarsfeld