We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves ${\mathcal C} \subset \mathbb P^r_K$ of maximal regularity with $°{\mathcal C} \leq 2r -3.$ In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces $X \subset \mathbb P^r_K$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for "an early decent of the Hartshorne-Rao function" of such surfaces. We use this criterion to give a lower bound on the degree for a class of these surfaces. Then, we study surfaces $X \subset \mathbb P^r_K$ for which $h^1(\mathbb P^r_K, {\mathcal I}_X(1))$ takes a value close to the possible maximum $°X - r +1.$ We give a lower bound on the degree of such surfaces. We illustrate our results by a number of examples, computed by means of {\sc Singular}, which show a rich variety of occuring phenomena.
Suppose X is a projective variety, which needs not be smooth, and L an ample divisor on X. We show that there are integers c and b such that for any nonnegative integer p, L^d is normally generated and embeds X as a variety who defining ideal has linear syzygies upto the p-th step (i.e. L^d has property N_p) for all d >= cp + b.
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 := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
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 a…
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebrai…
The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.
Our goal is to study the syzygies of the projective embeddings defined by ample line bundles on a fake projective plane S. The syzygies are studied in terms of the property $N_p$. For various kinds of ample line bundles, we give explicit lower bounds for their powers above which the property $N_p$ is satisfied.
We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface $V$ with isolated singularities and the Torelli properties of $V$ (in the sense of Dolgachev-Kapranov). We show in particular that hypersurfaces with a small Tjurina numbers are Torelli in this sense. When $V$ is a plane curve, or more interestingly, a surface in $P^3$, we discuss the stability of the reflexive sheaf of logarithmic vector fields along $V$. A new lower bound for the minimal degree of a syzygy associated to a 1-dimensional complete intersection is also g…
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, O_X) = 0), we obtain a numerical criterion for K+L to have property Np. When X is a regular variety of arbitrary dimension, under a mild condition, we give an explicit calculation for the regularity of the ideal sheaves of such embeddings.
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 $\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$ embedded by $\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.
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 asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.
Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m\geqslant n+1+p$. We also show the graded ring of sections $R(X, K_X+mA)$ is Koszul for $m\geqslant n+2$.
For an algebraic set $X$ (union of varieties) embedded in projective space, we say that $X$ satisfies property $\textbf{N}_{d,p}$, $(d\ge 2)$ if the $i$-th syzygies of the homogeneous coordinate ring are generated by elements of degree $< d+i$ for $0\le i\le p$ (see \cite{EGHP2} for details). Much attention has been paid to linear syzygies of quadratic schemes $(d=2)$ and their geometric interpretations (cf. \cite{AK},\cite{EGHP1},\cite{HK},\cite{GL2},\cite{KP}). However, not very much is actually known about the case satisfying property $\textbf{N}_{3,p}$. In this paper, we give a sharp upper…
The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.
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 misleadin…
In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences are given in terms of intersections with arbitrary linear subspaces. We use our results to bound homological invariants of some well-known projective varieties, to give a combinatorial characterization of quadratic monomial ideals with a long strand of linear syzygies, etc
The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre--Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology gro…
Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give a criterion for an integer m to guarantee that mP is normal.
Let $X \subset ¶^r$ be a nondegenerate projective variety and let $ν_{\ell} : ¶^r \to ¶^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective embedding $ν_{\ell} (X) \subset ¶^N$. We obtain that for a very ample line bundle $L \in {Pic}X$ such that $X \subset ¶H^0 (X,L)$ is $m$-regular in the sense of Castelnuovo-Mumford, $(X,L^{\ell})$ satisfies property $N_{\ell}$ for all $\ell \geq m$ (Theorem \ref{thm:main1}). This is a generalization of M. Green's work that $(¶^r,\mathcal{O}_{¶^r} (\ell))$ satisfi…
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.
We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.
In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfie…
This paper studies syzygies of curves that have been embedded in projective space by line bundles of large degree. The proofs take advantage of the relationship between syzygies and spaces of section of vector bundles associated to the given line bundles.
We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.
Let $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the syzygies with a generic first component vanishing on the singular locus subscheme of $V$. This yields in particular a positive answer in this setting to a question raised by Morihiko Saito and the first author. Finally we explain how our result can be used to improve the listing of Jacobian syzygies of a given degree by a computer algebra system such as Singular,…
We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening…
This work consists of two parts. In the first part we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces. From more general results we obtain in particular the following: a) Mukai's conjecture (and stronger variants of it) regarding projective normality and normal presentation for surfaces with Kodaira dimension 0, and uniform bounds for higher syzygies associated to adjoint linear series, b) effective bounds along the lines of Mukai's conjecture regarding proj…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend previous results in the literature.
We show that the implicit equation of a surface in 3-dimensional projective space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a given positive integer d, can be represented and computed from the linear syzygies of its parametrization if the base points are isolated and form locally a complete intersection.
Let $X$ be a reduced closed subscheme in $\mathbb P^n$. As a slight generalization of property $\textbf{N}_p$ due to Green-Lazarsfeld, we can say that $X$ satisfies property $\textbf{N}_{2,p}$ scheme-theoretically if there is an ideal $I$ generating the ideal sheaf $\mathcal I_{X/¶^n}$ such that $I$ is generated by quadrics and there are only linear syzygies up to $p$-th step (cf. \cite{EGHP1}, \cite{EGHP2}, \cite{V}). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property $\textbf{N}_{2,p}$(cf. \cite{CKP}, \cite{EGHP1}, \cite{EGHP2} \cite …
Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. There are also investigations about the shifts and the dimension of Betti numbers.
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded m…
The main result of this paper is the following: let d be a natural number >= 3; the line bundle O(1,...,1) on the product of d copies of the projective line P^1 satisfies Property N_p of Green-Lazarsfeld if and only if p <= 3.
We prove a duality theorem for simplicial complexes arising from a combinatorial construction we define, which applies to the squarefree monomial complexes for Veronese ideals of projective spaces and weighted projective spaces. Our theorem yields a formula for the multigraded Betti numbers of these Veronese ideals in terms of the reduced homology groups of these complexes which is dual to one given by Bruns and Herzog. We apply this formula in several ways, including by giving an algorithm for finding the highest syzygy of such a Veronese ideal.
We consider the graded space $R$ of syzygies for the coordinate algebra $A$ of projective variety $X=G/P$ embedded into projective space as an orbit of the highest weight vector of an irreducible representation of semisimple complex Lie group $G$. We show that $R$ is isomorphic to the Lie algebra cohomology $H=H^\bdot(\Lt,\CC)$, where $\Lt$ is graded Lie subalgebra of the graded Lie s-algebra $L=L_1\oplus\Lt$ Koszul dual to $A$. We prove that the isomorphism identifies the natural associative algebra structures on $R$ and $H$ coming from their Koszul and Chevalley DGA resolutions respectively.…
We describe the equations and Gröbner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Gröbner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.
The present paper is related to a conjecture made by Green and Lazarsfeld concerning 1-linear syzygies of curves embedded by complete linear systems of sufficiently large degrees. Given a smooth, irreducible, complex, projective curve $X$, we prove that the least integer $q$ for which the property $(M_q)$ fails for a line bundle $L$ on $X$ does not depend on $L$ as soon as its degree becomes sufficiently large. Consequently, this number is an invariant of the curve, and the statement of Green-Lazarsfeld's conjecture is equivalent to saying that this invariant equals the gonality of the curve. …
Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.
We give simple geometric proofs of Aprodu-Farkas-Papadima-Raicu-Weyman's theorem on syzygies of tangent developable surfaces of rational normal curves and Raicu-Sam's result on syzygies of K3 carpets. As a consequence, we obtain a quick proof of Green's conjecture for general curves of genus $g$ over an algebraically closed field $\mathbf{k}$ with $\operatorname{char}(\mathbf{k}) = 0$ or $\operatorname{char}(\mathbf{k}) \geq \lfloor (g-1)/2 \rfloor$. We also show the arithmetic normality of tangent developable surfaces of arbitrary smooth projective curves of large degree.
The Betti numbers of a graded module over the polynomial ring form a table of numerical invariants that refines the Hilbert polynomial. A sequence of papers sparked by conjectures of Boij and Söderberg have led to the characterization of the possible Betti tables up to rational multiples---that is, to the rational cone generated by the Betti tables. We will summarize this work by describing the cone and the closely related cone of cohomology tables of vector bundles on projective space, and we will give new, simpler proofs of some of the main results. We also explain some of the applications o…
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 determine the $r$ for which $L_r$ has $N_p$ property. In corollaries, we will relate the bounds on $r$ to the regularity of $B$. Examples in the last section show that several results are optimal.
On a special line bundle $L$ on a projective curve $C$ we introduce a geometric condition called $(Δ_q)$. When $L=K_C$ this condition implies gon$(C) \ge q+2$. For an arbitrary special $L$ we show that $(Δ_3)$ implies that $L$ has the well-known property $(M_3)$, generalizing a similar result proved by Voisin in the case $L=K_C$.
We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of …
First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …
We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.
