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 of the theory, including the one that originally motivated the conjectures of Boij and Söderberg, a proof of the Multiplicity Conjecture of Herzog, Huneke and Srinivasan.
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…
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…
We provide a number of new conjectures and questions concerning the syzygies of $\mathbb{P}^1\times \mathbb{P}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $\mathbb{P}^1\times \mathbb{P}^1$. These computations utilize linear algebra over finite fields and high-performance computing.
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…
In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves $C$ of genus 9. This includes a collection of all possible Betti tables for $C$. Moreover a direct correspondence between these Betti tables and the number and types of special linear series on $C$ is given. Especially for $\operatorname{Cliff}(C)=3$ the curve $C$ is contained in determinantal surface $Y$ on a 4-dimensional rational normal scroll $X\subset \mathbb{P}^8$ constructed from a base point free pencil of divisors of degree 5.
Hierarchical structure of graded Betti numbers in the quadratic strand
Korea Institute for Advanced Study (KIAS)（韩国高等研究院）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；Pukyong National University（釜庆国立大学）
We study the minimal free resolution of the Veronese modules of the polynomial ring in n variables, by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We characterize when they are Cohen-Macaulay and we give a sufficient condition for the linearity of their minimal free resolution. We also conjecture that in 2 variables the Veronese modules have always pure resolutions. In addition, we give a closed formula for their Hilbert series. As an application of our results, we calculate the complete Betti diagrams of the Veronese subrin…
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.
We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner bases for each higher syzygy module. In each case the given minimal Gröbner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied b…
In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.
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…
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor$, then the $2n$th Betti number of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.
Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
The School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
We show that for maximal Cohen-Macaulay modules over a homogeneous coordinate rings of smooth Calabi-Yau varieties $X$ computation of Betti numbers can be reduced to computations of dimensions of certain $\operatorname{Hom}$ groups in the bounded derived category $D^b(X)$. In the simplest case of a smooth elliptic curve $E$ imbedded into $\mathbb{P}^2$ as a smooth cubic we use our formula to get explicit answers for Betti numbers. Description of the automorphism group of the derived category $D^b(E)$ in terms of the spherical twist functors of Seidel and Thomas plays a major role in our approa…
This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on the projective plane P2 and its blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up. These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for P2 follow from those for the blow-up by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions.
Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of the local cohomology of Tor_i(A,B) under the assumption that the Krull dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies, Groebner bases, products and powers of ideals, and to the relationship of the Rees and Symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first (roughly) (n-1)/2 steps of the resolution of I are linear, then I^…
Let $M(d,χ)$ with $(d,χ)=1$ be the moduli space of semistable sheaves on $\mathbb{P}^2$ supported on curves of degree $d$ and with Euler characteristic $χ$. The cohomology ring $H^*(M(d,χ),\mathbb{Z})$ of $M(d,χ)$ is isomorphic to its Chow ring $A^*(M(d,χ))$ by Markman's result. W. Pi and J. Shen have described a minimal generating set of $A^*(M(d,χ))$ consisting of $3d-7$ generators, which they also showed to have no relation in $A^{\geq d-2}(M(d,χ))$. We compute the two Betti numbers $b_{2(d-1)}$ and $b_{2d}$ of $M(d,χ)$ and as a corollary we show that the generators given by Pi-Shen have no…
Let X be a projective scheme; let M and N be two coherent O_X-modules. Given an integer m, we present an algorithm for computing the global extension module Ext^m(X;M,N). In particular, this allows one to compute the sheaf cohomology H^m(X,N) and to construct the sheaf corresponding to an element of the module Ext^1(X;M,N). This algorithm can be implemented using only the computation of Grobner bases ans syzygies, and it has been implemented in the computer algebra system Macaulay2.
This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli…
Let $X$ be a smooth projective curve, $G$ a reductive group, and $Bun_G(X)$ the moduli of $G$-bundles on $X$. For each point of $X$, the Satake category acts by Hecke modifications on sheaves on $Bun_G(X)$. We show that, for sheaves with nilpotent singular support, the action is locally constant with respect to the point of $X$. This equips sheaves with nilpotent singular support with a module structure over perfect complexes on the Betti moduli $Loc_{G^\vee}(X)$ of dual group local systems. In particular, we establish the "automorphic to Galois" direction in the Betti Geometric Langlands corr…
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 …
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 …
We show that a moduli space of slope-stable sheaves over a K3 surface is an irreducible hyperkähler manifold if and only if its second Betti number is the sum of its Hodge numbers $h^{2,0}$, $h^{1,1}$ and $h^{0,2}$.
We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.
We compute Betti numbers of the moduli spaces of arbitrary rank stable sheaves on ruled surfaces. Our result generalizes the formula of Goettsche for rank one sheaves and the formula of Yoshioka for rank two sheaves. It also confirms the conjecture of Manschot for arbitrary rank sheaves on the Hirzebruch surfaces.
On leading Gopakumar--Vafa invariants of local $\mathbb{P}^2$ and leading Betti numbers of moduli spaces of one-dimensional sheaves on $\mathbb{P}^2$
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Institute for Math and AI, Wuhan University（武汉大学数学与人工智能中心）
Boij-Söderberg theory characterizes syzygies of graded modules and sheaves on projective space. This paper continues earlier work with S. Sam, extending the theory to the setting of $GL_k$-equivariant modules and sheaves on Grassmannians. Algebraically, we study modules over a polynomial ring in $k n$ variables, thought of as the entries of a $k \times n$ matrix. We give equivariant analogues of two important features of the ordinary theory: the Herzog-Kühl equations and the pairing between Betti and cohomology tables. As a necessary step, we also extend previous results, concerning the base c…
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma-Tibăr in the case of hypersurfaces with a $1$-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof…
We calculate Betti numbers of the framed moduli space of instantons on $\hat{{\bf C}^2/{\bf Z}_2}$, under the assumption that the corresponding torsion free sheaves $E$ have vanishing properties ($Hom(E,E(-l_\infty))=Ext^2(E,E(-l_\infty))=0$). Moreover we derive the generating function of Betti numbers and obtain closed formulas. On the other hand, we derive a universal relation between the generating function of Betti numbers of the moduli spaces of stable sheaves on $X$ with an $A_1$-singularity and that on $\hat{X}$ blow-uped at the singularity, by using Weil conjecture. We call this the $O…
Let $\mathbf{M}_d$ be the moduli space of stable sheaves on $\mathbb{P}^2$ with Hilbert polynomial $dm+1$. In this paper, we determine the effective and the nef cone of the space $\mathbf{M}_d$ by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space $\mathbf{M}_6$. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction i…
This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. T…
We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially. More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many exa…
We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points…
We study certain moduli spaces of sheaves on Enriques surfaces thereby obtaining, in every odd dimension, new examples of Calabi-Yau manifolds. We describe the geometry (canonical bundle, fundamental group, second Betti number and certain Hodge numbers) of these moduli spaces showing, in partial analogy to the well-known case of sheaves on K3 or Abelian surfaces, how the geometry of the surface reflects that of the moduli space itself.
We introduce and survey a Betti form of the geometric Langlands conjecture, parallel to the de Rham form developed by Beilinson-Drinfeld and Arinkin-Gaitsgory, and the Dolbeault form of Donagi-Pantev, and inspired by the work of Kapustin-Witten in supersymmetric gauge theory. The conjecture proposes an automorphic category associated to a compact Riemann surface X and complex reductive group G is equivalent to a spectral category associated to the underlying topological surface S and Langlands dual group G^. The automorphic category consists of suitable C-sheaves on the moduli stack Bun_G(X) o…
Inspired by some recent work of M. Farber, W. Lück and M. Shubin on L2 homotopy invariants of infinite Galois coverings of simplicial complexes (L2 Betti numbers and Novikov-Shubin invariants), this article extends Atiyah's L2 index theory to coherent analytic sheaves on complex analytic spaces. Let $X$ be a complex analytic space with a proper cocompact biholomorphic action of a discrete group $G$. Let $F$ be a $G$-equivariant coherent analytic sheaf on $X$. We give a meaningful notion of a L2 section of $F$ on $X$. We also construct L2 cohomology groups. We prove that these L2 cohomology gro…
