Graded Betti Numbers Research Articles

Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes

In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set X in a projective space Pn. More concretely, via Generic initial ideals (Gins) method we mainly consider ‘tailing’ Betti numbers, whose homological index is at least codim(X,Pn). For this purpose, we first introduce a key definition ‘ND(1) property’, which provides a suitable ground where one can generalize the concepts such as ‘being nondegenerate’ or ‘of minimal degree’ from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing Betti numbers. Next, we recall basic notions and facts on Gins theory and we analyze the generation structure of the reverse lexicographic (rlex) Gins of 3-regular ND(1) subschemes. As a result, we present exact formulae for these tailing Betti numbers, which connect them with linear normality of general linear sections of X∩Λ with a linear subspace Λ of dimension at least codim(X,Pn). Finally, we consider some applications and related examples.

A Proof of Evans’ Convexity Conjecture

Given a sequence of degrees 𝔸 = {a 1,…, a n } with 1 ≤ a 1 ≤ a 2 ≤ … ≤a n , an ideal L is called an 𝔸 lex-plus-powers ideal if L minimally contains and is otherwise as lex as possible. Such ideals are conjectured by Eisenbud, Green, and Harris, to exhibit extremal Hilbert function growth (among all ideals also containing a regular sequence in degrees 𝔸), and by Charalambous and Evans to exhibit maximal graded Betti numbers (among all ideals containing a regular sequence in degrees 𝔸 and attaining the same Hilbert function). Of course, given a sequence of degrees 𝔸 and a Hilbert function ℋ, an 𝔸 lex-plus-powers ideal with Hilbert function ℋ may or may not exist. Evans has conjectured that the existence of such ideals is governed by a certain convexity property: that is, given a Hilbert function ℋ and three sequences of degrees 𝔸 ≤ 𝔹 ≤ 𝕃 where 𝕃 gives the degrees of the pure powers in the lex ideal attaining ℋ, if there exists an 𝔸 lex-plus-powers ideal attaining ℋ, then there exists a 𝔹 lex-plus-powers ideal attaining ℋ. In this paper we give a proof of Evans’ Convexity Conjecture. As an application, we use Convexity to show that the conjectures of Eisenbud, Green, and Harris, and Charalambous and Evans are equivalent to similar statements utilizing a definition of lex-plus-powers ideals that does not require minimal containment of .

Cover Product and Betti Polynomial of Graphs

AbstractThe cover product of disjoint graphs G and H with fixed vertex covers C(G) and C(H), is the graphwith vertex set V(G) ∪ V(H) and edge setWe describe the graded Betti numbers of GeH in terms of those of. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph G such that reg R/I(G) = μS(G) + k, where, I(G) denotes the edge ideal of G, reg R/I(G) is the Castelnuovo–Mumford regularity of R/I(G) and μS(G) is the induced or strong matching number of G; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The h-vector of R/I(G e H) is described in terms of the h-vectors of R/I(G) and R/I(H). Furthermore, in a diòerent direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

Tight combinatorial manifolds and graded Betti numbers

In this paper, we study the conjecture of Kuhnel and Lutz, who state that a combinatorial triangulation of the product of two spheres $$\mathbb S^i \times \mathbb S^j$$ with $$j \ge i$$ is tight if and only if it has exactly $$i+2j+4$$ vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when $$j>2i$$ and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

The Tangent Cone of a Local Ring of Codimension 2

Let \((S, \mathfrak {n})\) be a regular local ring and let \(I \subseteq \mathfrak {n}^{2} \) be a perfect ideal of S. Sharp upper bounds on the minimal number of generators of I are known in terms of the Hilbert function of R = S/I. Starting from information on the ideal I, for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal I∗ which defines the tangent cone of R or to give information on its graded structure. Motivated by papers of S. C. Kothari and S. Goto et al. concerning the leading ideal of a complete intersection I = (f, g) in a regular local ring, we present results provided ht (I) = 2. If I is a complete intersection, we prove that the Hilbert function of R determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if ν(I) > 2 and a careful investigation can be provided when ν(I) = 3. Several examples illustrating our results are given.

A combinatorial interpretation for Schreyer's tetragonal invariants

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers b_1 and b_2 , associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width.

On syzygies, degree, and geometric properties of projective schemes with property [formula omitted

Let X be a reduced, but not necessarily irreducible closed subscheme of codimension e in a projective space. One says that X satisfies property Nd,p(d≥2) if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree <d+i for 0≤i≤p (see [10] for details). Much attention has been paid to linear syzygies of quadratic schemes (d=2) and their geometric interpretations (cf. [1,9,15–17]). However, not very much is actually known about algebraic sets satisfying property Nd,p, d≥3. Assuming property Nd,e, we give a sharp upper bound deg⁡(X)≤(e+d−1d−1). It is natural to ask whether deg⁡(X)=(e+d−1d−1) implies that X is arithmetically Cohen–Macaulay (ACM) with a d-linear resolution. In case of d=3, by using the elimination mapping cone sequence and the generic initial ideal theory, we show that deg⁡(X)=(e+22) if and only if X is ACM with a 3-linear resolution. This is a generalization of the results of Eisenbud et al. (d=2) [9,10].We also give more general inequality concerning the length of the finite intersection of X with a linear space of not necessary complementary dimension in terms of graded Betti numbers. Concrete examples are given to explain our results.

Betti tables of reducible algebraic curves

We study the Betti tables of reducible algebraic curves with a focus on connected line arrangements and provide a general formula for computing the quadratic strand of the Betti table for line arrangements that satisfy certain hypotheses. We also give explicit formulas for the entries of the Betti tables for all curves of genus zero and one. Last, we give formulas for the graded Betti numbers for a class of curves of higher genus.

Sharp bounds for higher linear syzygies and classifications of projective varieties

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 algebraic set $X$ and those of its inner projection $X_q$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).

On the Koszul property of toric face rings

Toric face rings are a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e and Herzog et al., graded Betti numbers of $k$ over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by R\"{o}mer about the sufficient condition for the Koszul property. The conjecture is inspired by Fr\"{o}berg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.

Geometry of Wachspress surfaces

Let Pd be a convex polygon with d vertices. The associated Wach- spress surface Wd is a fundamental object in approximation theory, defined as the image of the rational map P 2 wd ! P d 1 , determined by the Wachspress barycentric coordinates for Pd. We show wd is a regular map on a blowup Xd of P 2 and if d > 4 is given by a very ample divisor on Xd, so has a smooth image Wd. We determine generators for the ideal of Wd, and prove that in graded lex order, the initial ideal of IWd is given by a Stanley-Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen-Macaulay, of Castelnuovo-Mumford regularity two, and determine all the graded betti numbers of IWd .

Poset embeddings of Hilbert functions and Betti numbers

We study inequalities between graded Betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps, defined earlier by us in Caviglia and Kummini (2013) [5]. We show that if graded Betti numbers do not decrease when we replace ideals in an algebra by their embedded versions, then the same behaviour is carried over to ring extensions. As a corollary we give alternative inductive proofs of earlier results of Bigatti, Hulett, Pardue, Mermin, Peeva, Stillman and Murai. We extend a hypersurface restriction theorem of Herzog and Popescu to the situation of embeddings. We show that we can obtain the Betti table of an ideal in the extension ring from the Betti table of its embedded version by a sequence of consecutive cancellations. We further show that the lex-plus-powers conjecture of Evans reduces to the Artinian situation.

Resolutions of 2 and 3 Dimensional Rings of Invariants for Cyclic Groups

Let G be the cyclic group of order n, and suppose F is a field containing a primitive nth root of unity. We consider the ring of invariants F[W] G of a three dimensional representation W of G where G ⊂ SL(W). We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of F[W] G . The case where W is any two dimensional representation of G is also handled.

Dependence of Betti Numbers on Characteristic

We study the dependence of graded Betti numbers of monomial ideals on the characteristic of the base field. The examples we describe include bipartite ideals and Stanley–Reisner ideals of vertex-decomposable complexes. We give a description of bipartite graphs and, using discrete Morse theory, provide a way of looking at the homology of arbitrary simplicial complexes through bipartite ideals.

Regularity 3 in edge ideals associated to bipartite graphs

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree >i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number β i,i+4 in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.

On projective curves of next to maximal regularity

It is known that if a rational curve C⊂Pr (r≥4) of degree d≥r+4 fails to be (d−r)-regular then C admits a unique (d−r+1)-secant line L and the arithmetic genus of C is at most 1. In this paper, we study the effect of such a secant line on algebraic and geometric properties of the curve C. We show that if the singular locus of C does not lie on L then C is obtained by a simple linear projection of a curve C˜⊂Pr+1 of maximal regularity. Also we show that if d≤2r−2 then C∪L is 3-regular which enables us to estimate the Hartshorne–Rao module and the graded Betti numbers of C.

On The Binomial Edge Ideal of a Pair of Graphs

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

Poincaré Series of some Hypergraph Algebras

A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,\ldots, x_n]/(x_{i_1}\cdots x_{i_k}; \{i_1,\ldots,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We determine the Poincaré series $P_{R_H}(t)=\sum_{i=1}^\infty\dim_k\mathrm{Tor}_i^{R_H}(k,k)t^i$ for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincaré series of the graph algebra of the wheel graph.

Linear quotients of Artinian Weak Lefschetz algebras

We study the Hilbert function and the graded Betti numbers for “generic” linear quotients of Artinian standard graded algebras, especially in the case of Weak Lefschetz algebras. Moreover, we investigate a particular property of Weak Lefschetz algebras, the Betti Weak Lefschetz Property, which makes possible to completely determine the graded Betti numbers of a generic linear quotient of such algebras.

Scheme-Theoretic Complete Intersections in ℙ1 × ℙ1

The zero-dimensional subschemes of 𝒬 = ℙ1 × ℙ1 arising as scheme-theoretic complete intersections of two curves are considered. The main goal is to describe the possible Hilbert functions and to give some information on the graded Betti numbers of such schemes.