arXiv subjects
Publications and source records attributed to Robert Lazarsfeld.
This note points out a gap in the proof of one of the technical results in the paper "Asymptotic Invariants of Base Loci", that appeared in Ann. Inst. Fourier (Grenoble) 56 (2006), 1701-1734. We provide a correct proof of this result.
Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic curve.
A Steiner bundle is a vector bundle on projective space arising as the cokernel of the map defined by a matrix of linear forms. These come up in various geometric settings, and by now they are the subject of a considerable literature. Starting with work of Dolgachev and Kapranov from 1993, several authors have considered the question of whether one can recover from the bundle the geometric data used to construct it. Here we prove such Torelli-type statements for the tautological bundles associated to sufficiently positive divisors on any very ample linear series. In an appendix, we give a new proof of the result of Dolgachev-Kapranov.
By analyzing degeneracy loci over projectivized vector bundles, we recompute the degree of the discriminant locus of a vector bundle and provide a new proof of the Bogomolov instability theorem.
This paper begins the exploration of what we call measures of association between two irreducible complex projective varieties of the same dimension. The idea is to study from various points of view the minimal complexity of correspondences between them. We extend to this setting results about degrees of irrationality for hypersurfaces, and we study joint covering invariants for pairs of curves and hypersurfaces. We also propose a number of conjectures and open problems.
We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macualay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.
Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a splitting hypothesis that in practice is rarely satisfied. We show that multiplier ideals lead to a quite clean statement in the case of excess vanishing. Along the way, we give simplified and somewhat strengthened accounts of results of Tan-Viehweg and Sun.
This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu and Weyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of a rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of a general canonical curve. The present note, which is the write-up of a talk given by the second author at the Current Events seminar at the 2019 JMM, surveys this circle of ideas.
The Konno invariant of a projective variety X is the minimum geometric genus of the fiber of a rational pencil on X. It was computed by Konno for surfaces in P^3, and in general can be viewed as a measure of the complexity of X. We estimate Konno(X) for some natural classes of varieties, including sharp asymptotics for polarized K3 surfaces. In an appendix, we give a quick proof of a classical formula due to Deligne and Hoskin for the colength of an integrally closed ideal on a surface.
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.
We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if X is a very general smooth hypersurface of dimension n and degree d \ge 2n+1, then any dominant rational mapping from X to projective n-space must have degree at least d-1. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti-Eisenbud.
Inspired by the methods of Voisin, the first two authors recently proved that one could read off the gonality of a curve C from the syzygies of its ideal in any one embedding of sufficiently large degree. This was deduced from from a vanishing theorem for the asymptotic syzygies associated to an arbitrary line bundle B on C. The present paper extends this vanishing theorem to a smooth projective variety X of arbitrary dimension. Specifically, given a line bundle B on X, we prove that if B is p-jet very ample (i.e. the sections of B separate jets of total weight p+1) then the weight one Koszul cohomology group K_{p,1}(X, B; L) vanishes for all sufficiently positive L. In the other direction, we show that if there is a reduced cycle of length p+1 that fails to impose independent conditions on sections of B, then the Koszul group in question is non-zero for very positive L.
We give a quick new approach to the main cases of the nonvanishing theorems of first and third authors concerning the asymptotic behavior of the syzygies of a projective variety as the positivity of the embedding line bundle grows. Specifically, we present a surprisingly elementary and concrete proof of the asymptotic nonvanishing of Veronese syzygies, and we obtain effective results for arithmetically Cohen-Macaulay varieties. The idea is that one can reduce the statements to some simple computations with monomials.
This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results and conjectures about the asymptotics of Betti numbers. Finally we discuss the case of syzygies of weight one, and the gonality conjecture on the syzygies of curves of large degree. The exposition also discusses numerous open questions and conjectures.
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.
Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an algebraically closed field, then M_L is slope-stable for any sufficiently positive L.
The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of the embedding line bundle grows. Then, in an attempt to render the conjecture plausible, we prove a result suggesting that this is in any event the typical behavior from a probabilistic point of view. Specifically, we consider a "random" Betti table with a fixed number of rows, sampled according to a uniform choice of Boij-Soderberg coefficients. We compute the asymptotics of the entries as the length of the table goes to infinity, and show that they become normally distributed with high probability.
This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleading. For Veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. Finally, we propose a number of questions and open problems concerning asymptotic syzygies of higher-dimensional varieties.