Robert Lazarsfeld

On a theorem of Castelnuovo, and the equations defining space curves

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 w Notat ion and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 w A Regularity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 w A Rationality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 w The Existence of Secant Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 w Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville

Introduction 389 w Notation and conventions 392 w Deformations of cohomology groups 392 w Generic vanishing criteria for topologically trivial line bundles 397 w A Nakano-type generic vanishing theorem 401 w An application to surfaces 405 References 407

Uniform bounds and symbolic powers on smooth varieties

The purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety. In particular, weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety.Let

Higher obstructions to deforming cohomology groups of line bundles

Our purpose is to study the cohomological properties of topologically trivial holomorphic line bundles on a compact Kahler manifold. (See [GLl, GL2] for our prior work on this topic.) Let M be a compact connected Kahler manifold and, as usual, let Pico(M) denote the complex torus parametrizing isomorphism classes of topologically trivial holomorphic line bundles on M. We are interested in the analytic subvarieties Si(M) , S~(M) ~ Pico(M) defined by

On the connectedness of degeneracy loci and special divisors

Introduction Let C be a smooth complex projective curve of genus g, and let J be the Jacobian of C. Upon choosing a base-point in C, J may be identified with the set of linear equivalence classes of divisors of degree d on C. Denote by W~ the algebraic subvariety of J parametrizing divisors which move in a linear system of dimension at least r. A fundamental theorem of Kempf [9] and Kleiman and Laksov [11, 12] asserts tha t these loci are nonempty when their expected dimension

Singularities of theta divisors and the birational geometry of irregular varieties

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we prove a conjecture of Arbarello and DeConcini characterizing principally polarized abelian varieties whose theta divisors are singular in codimension one. Secondly, we study the holomorphic Euler characteristics of varieties of general type having maximal Albanese dimension: we verify a conjecture of Kollar for subvarieties of abelian varieties, but show that it fails in general. Finally, we give a surprisingly simple new proof of a fundamental theorem of Kawamata concerning the Albanese mapping of projective varieties of Kodaira dimension zero.

Global generation of pluricanonical and adjoint linear series on smooth projective threefolds

The purpose of this paper is to show how the cohomological techniques developed by Kawamata, Reid, Shokurov, and others lead to some effective and practical results of Reider-type on freeness of linear series on smooth complex projective threefolds. In recent years, there has been a great deal of interest in the geometric properties of pluricanonical and adjoint linear systems on surfaces and higherdimensional varieties. Among other things, one wants to understand as explicitly as possible when the systems in question are base-point free or very ample. Modem work in this area goes back to Kodaira [Kod] and Bombieri [Bmb], who studied pluricanonical maps of surfaces of general type. More recently, many of their results have emerged as special cases of a celebrated theorem of Reider [Rdr]. Reider uses vector bundle techniques to show that, if B is a nef line bundle on a surface X such that B2 > 5, then KX + B is globally generated unless there exist certain special curves C c X such that B * C < 1; he also obtains analogous criteria for KX + B to be very ample. A cohomological approach to these theorems, based on Miyaokas vanishing theorem for Zariski decompositions, was given by Sakai [Sak2].

Restricted volumes and base loci of linear series

We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.

Jumping coefficients of multiplier ideals

We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser and Vaquie imply that if \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to show prove that this holds also when the singular locus of f has positive dimension. In a more algebraic direction, we show that the number of such jumping coefficients bounds the uniform Artin-Rees number of the principal ideal (f) in the sense of Huneke: in the case of isolated singularities, this in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along the way, we establish a general result relating multiplier to Jacobian ideals. We also explore the extension of these ideas to the setting of graded families of ideals. The paper contains many concrete examples.

Contact loci in arc spaces

We study loci of arcs on a smooth variety defined by order of contact with a fixed subscheme. Specifically, we establish a Nash-type correspondence showing that the irreducible components of these loci arise from (intersections of) exceptional divisors in a resolution of singularities. We show also that these loci account for all the valuations determined by irreducible cylinders in the arc space. Along the way, we recover in an elementary fashion-without using motivic integration-results of the third author relating singularities to arc spaces. Moreover, we extend these results to give a jet-theoretic interpretation of multiplier ideals.