Related papers: Asymptotic syzygies of algebraic varieties

We establish precise nonvanishing results for asymptotic syzygies of smooth projective varieties. This refines Ein-Lazarsfeld's asymptotic nonvanishing theorem. Combining with the author's previous asymptotic vanishing result, we completely…

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…

The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of…

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.…

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of…

Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L :=…

The purpose of this paper is to establish an effective non-vanishing theorem for the syzygies of an adjoint-type line bundle on a smooth variety, as the positivity of the embedding increases. Our purpose here is to show that for an adjoint…

Let $X \subset \P^r$ be a nondegenerate projective variety and let $\nu_{\ell} : \P^r \to \P^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective…

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,…

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=\P_C (\mathcal{E})$ with projection map $\pi:X\to C$. In this paper we investigate the minimal free resolution of homogeneous coordinate…

We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow…

We show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds when p is small, by studying the Bridgeland-King-Reid-Haiman…

Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X,…

The purpose of the paper is to give a sharp asymptotic description of the weights that appear in the syzygies of a toric variety. We prove that as the positivity of the embedding increases, in any strand of syzygies, torus weights after…

Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we…

We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois…

We prove that the minimal free resolution of the secant variety of a curve is asymptotically pure. As a corollary, we show that the Betti numbers of converge to a normal distribution.

By adapting methods of Ein-Erman-Lazarsfeld, we prove an analogue of the Ein-Lazarsfeld result on asymptotic syzygies for Veronese embeddings, in the setting of weighted projective spaces of the form $\mathbb{P}(1^n,2)$.

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results…

Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford…