Graded Betti Numbers Research Articles

Characterization of the Graded Betti Numbers for Almost Complete Intersections

Using standard mapping cone procedure of the inclusion map of an ideal generated by a suitable regular sequence inside a Gorenstein ideal and by a deep analysis of possible cancellations, we give a numerical characterization of the graded Betti sequences admissible for almost complete intersections of height three.

Algebraic Properties of Product of Graphs

Let G and H be two simple graphs and let G*H denotes the graph theoretical product of G by H. In this article, we provide graded Betti numbers, Castelnuovo–Mumford regularity, projective dimension, h-vector and Hilbert series of G*H in terms of that information of G and H. More generally, we present explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. We will also prove that the family of graphs for which the regularity of each edge ring equals the maximum number of pairwise 3-disjoint edges, is closed under product of graphs.

The minimal resolution conjecture for points on del Pezzo surfaces

Mustaţa (1997) stated a generalized version of the minimal resolution conjecture for a set Z of general points in arbitrary projective varieties and he predicted the graded Betti numbers of the minimal free resolution of IZ. In this paper, we address this conjecture and we prove that it holds for a general set Z of points on any (not necessarily normal) del Pezzo surface X⊆ℙd — up to three sporadic cases — whose cardinality |Z| sits into the interval [PX(r−1),m(r)] or [n(r),PX(r)], r≥4, where PX(r) is the Hilbert polynomial of X, m(r):=12dr2+12r(2−d) and n(r):=12dr2+12r(d−2). As a corollary we prove: (1) Mustaţa’s conjecture for a general set of s≥19 points on any integral cubic surface in ℙ3; and (2) the ideal generation conjecture and the Cohen–Macaulay type conjecture for a general set of cardinality s≥6d+1 on a del Pezzo surface X⊆ℙd.

High rank linear syzygies on low rank quadrics

We study the linear syzygies of a homogeneous ideal $I \subseteq S = {\rm Sym}_{{\bf k}}(V)$, focussing on the graded betti numbers $$ b_{i,i+1} = {\rm dim}_{{\bf k}} {\rm Tor}_i (S/I,{\bf k})_{i+1}. $$ For a variety $X$ and divisor $D$ with $V = H^0(D)$, what conditions on $D$ ensure that $b_{i,i+1} \ne 0$? Eisenbud has shown that a decomposition $D \sim A +B$ such that $A$ and $B$ have at least two sections gives rise to determinantal equations (and corresponding syzygies) in $I_X$; and conjectured that if $I_2$ is generated by quadrics of rank $\le 4$, then the last nonvanishing $b_{i,i+1}$ is a consequence of such equations. We describe obstructions to this conjecture and prove a variant. The obstructions arise from toric specializations of the Rees algebra of Koszul cycles, and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank. This gives one answer to a question posed by Eisenbud and Koh.

Minimal free resolution for points on surfaces

A long-standing problem in Algebraic Geometry and Commutative Algebra is to determine the minimal graded free resolution of a 0-dimensional scheme Z in Pn or in an arbitrary projective variety X. In [18], M. Mustaţă (1998) predicted the graded Betti numbers of the minimal free resolution of a general set of distinct points Z in X. In this paper, we state a refined version of Mustaţăʼs conjecture (MRC) and we predict the existence of a non-empty open subset U⊂Hilbs(X) such that any [Z]∈U has a minimal graded free resolution without ghost terms (WMRC). In this paper, we are going to prove: (1) for any s⩾(d+33)−1 there exists a (d+22)-dimensional family of irreducible generically smooth surfaces of degree d in P3 satisfying the WMRC for s; and (2) any smooth cubic surface satisfies the MRC for any s⩾19.

The Hilbert Scheme of Buchsbaum space curves

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H * 0 (ℐ C ) and Rao module M:=H * 1 (ℐ C ). By taking suitable generizations (deformations to a more general curve) C ′ of C, we simplify the minimal free resolution of I(C) by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of I(C ′ ). Using this for Buchsbaum curves of diameter one (M v ≠0 for only one v), we establish a one-to-one correspondence between the set 𝒮 of irreducible components of H(d,g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of 𝒮), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.

Path ideals of rooted trees and their graded Betti numbers

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal I t ( Γ ) is generated by monomials that correspond to directed paths of length ( t − 1 ) in Γ. In this paper, we study algebraic properties and invariants of I t ( Γ ) . We give a recursive formula to compute the graded Betti numbers of I t ( Γ ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when I t ( Γ ) has a linear resolution.

On subschemes of $0$-dimensional schemes with given graded Betti numbers

On subschemes of $0$-dimensional schemes with given graded Betti numbers

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in P N , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N = 2 , we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I A defining A , generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.

Graded Betti Numbers of the Logarithmic Derivation Module

Let Q ∈ 𝕂[x 1,…, x n ] = S be a homogeneous polynomial of degree d. The freeness of the logarithmic derivation module, D(Q), and of its natural generalizations has been widely studied. In the free case, , where the d i 's are the exponents of the module; and as a direct consequence of the Saito–Ziegler criterion, the formula holds. In this article, we give a generalization of this formula in the non-free case. Moreover, we show that an equivalent formula is also true in the quasi-homogeneous case, and show to what extent it can be generalized for arbitrary polynomials.

Moduli Spaces of Reflexive Sheaves of Rank 2

AbstractLet ℱ be a coherent rank 2 sheaf on a scheme Y ⊂ ℙn of dimension at least two and let X ⊂ Y be the zero set of a section σ ∈ H0(ℱ). In this paper, we study the relationship between the functor that deforms the pair (ℱ, σ) and the two functors that deform F on Y, and X in Y, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of e.g., the moduli scheme MY(P) of stable sheaves on a threefold Y at (ℱ), and the scheme structure at (X) of the Hilbert scheme of curves on Y become closely related. Using this relationship, we get criteria for the dimension and smoothness of MY(P) at (ℱ), without assuming Ext2(ℱ,ℱ) = 0. For reflexive sheaves on Y = ℙ3 whose deficiencymodule M = H*1 (ℱ) satisfies 0Ext2(M,M) = 0 (e.g., of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H*0 (ℱ). Moreover, we show that every irreducible component of Mℙ3 (P) containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of Mℙ3 (P) that contains a reflexive stable sheaf with “small” deficiency module M.

Free resolutions of lex-ideals over a Koszul toric ring

In this paper, we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular, we study in which toric ring R all lex-ideals are componentwise linear. We give a certain necessity and sufficiency condition for this property, and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from the Segre product P×· · ·×P, every Hilbert function of a graded ideal is attained by a lex-ideal and that lex-ideals have the greatest graded Betti numbers among all ideals having the same Hilbert function.

Graded mapping cone theorem, multisecants and syzygies

Let X be a reduced closed subscheme in P n . As a slight generalization of property N p due to Green–Lazarsfeld, one says that X satisfies property N 2 , p scheme-theoretically if there is an ideal I generating the ideal sheaf I X / P n such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. Eisenbud et al. (2005) [8], Vermeire (2001) [20]). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N 2 , p (cf. Choi, Kwak, and Park (2008) [6], Eisenbud et al. (2005) [8], Kwak and Park (2005) [15]). In this case, the Castelnuovo regularity and normality can be obtained by the blowing-up method as reg ( X ) ⩽ e + 1 where e is the codimension of a smooth variety X (cf. Bertram, Ein, and Lazarsfeld (2003) [3]). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry (cf. Beheshti and Eisenbud (2010) [2], Kwak (1998) [14], Kwak and Park (2005) [15], Lazarsfeld (1987) [16]. We first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the non-complete embedding of X. For applications, we give an optimal bound on the length of zero-dimensional intersections of X and a linear space L in terms of graded Betti numbers. We also deduce several theorems about the relationship between X and its projections with respect to the geometry and syzygies for a projective scheme X satisfying property N 2 , p scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers of the projection for the case of N d , p , d ⩾ 2 , but also geometric structures for projections according to moving the center.

Combinatorial Rigidity of 3-dimensional Simplicial Polytopes

A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers, which are important invariants coming from combinatorial commutative algebra. We find a necessary condition to be combinatorially rigid for 3-dimensional reducible simplicial polytopes and provide some rigid reducible simplicial polytopes.

A Combinatorial Proof of a Formula for Betti Numbers of a Stacked Polytope

For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}({\bf k}[\Delta])$ of the Stanley-Reisner ring ${\bf k}[\Delta]$ over a field ${\bf k}$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $\beta_{k-1,k}({\bf k}[\Delta])=(k-1){n-d\choose k}$. We prove this combinatorially.

Stable Postulation and Stable Ideal Generation: Conjectures for Fat Points in the Plane

It is an open problem to determine the Hilbert function and graded Betti numbers for the ideal of a fat point subscheme supported at general points of the projective plane. In fact, there is not yet even a general explicit conjecture for the graded Betti numbers. Here we formulate explicit asymptotic conjectures for both problems. We work over an algebraically closed field $K$ of arbitrary characteristic.

Generic Initial Ideals, Graded Betti Numbers, and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are: 1. Let I be an ideal of R = K[x 1, x 2,…, x n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.

Consecutive cancellations in Betti numbers of local rings

Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay's Theorem there exists a lexicographic ideal L = Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R, n) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire's construction, we present several applications. This connection between the graded perspective and the local one is a new viewpoint, and we hope it will be useful for studying the numerical invariants of classes of local rings.

Syzygies of the secant variety of a curve

We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.

First Nonlinear Syzygies of Ideals Associated to Graphs

Consider an ideal I ⊂ K[x 1,…, x n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β s, s+3. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.