Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1512.06384

A vanishing theorem for weight one syzygies

Lawrence Ein·Robert Lazarsfeld·David Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.AGmath.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lawrence Ein, Robert Lazarsfeld, David Yang. 2016-09-21. A vanishing theorem for weight one syzygies. https://doi.org/10.2140/ant.2016.10.1965

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Moduli Stacks of $G$-Curves in Homotopy Theory at Height $p-1$

Let $p$ be odd and $G' = \mathbb{Z}/p \rtimes \mathbb{Z}/(p-1)^2$ the maximal finite subgroup of the Morava stabilizer group at height $p-1$. Inverse Galois theory produces from $G'$ alone a curve $X$, the unique curve of minimal genus with $\operatorname{Aut}(X) \simeq G'$; its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a $G'$-equivariant equivalence between the deformations of $X$ and Lubin--Tate space, so that the Lubin--Tate action of $G'$ is the action of $\operatorname{Aut}(X)$ on deformations of the curve. The proof is a coordinate-free Kodaira--Spencer argument reducing to a single character count. The action becomes explicit: $G'$ acts through $\mathbb{F}_p \rtimes \mathbb{F}_p^\times$ shifting and scaling $p+1$ points on $\mathbb{P}^1$. From this we compute $H^*(G', π_* E_{p-1})$ and its Tate cohomology. One identity, $π^{p-1} = -p$, runs through every section.

math.AG↗