Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO _Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ³. For almost all such schemes the graded Betti numbers in ℙ³ depend only on the Hilbert function ofX in ℙ³. Work done with financial support of M.U.R.S.T., while the authors were members of C.N.R.     

Reduced Gorenstein codimension three subschemes of projective space

It is known, from work of Diesel, which graded Betti numbers are possible for Artinian Gorenstein height three ideals. In this paper we show that any such set of graded Betti numbers in fact occurs for a reduced set of points in , a stick figure in , or more generally, a good linear configuration in . Consequently, any Gorenstein codimension three scheme specializes to such a ``nice' configuration, preserving the graded Betti numbers in the process. This is the codimension three Gorenstein analog of a classical result of arithmetically Cohen-Macaulay codimension two schemes.

     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Level Algebras,Lex Segments,and Minimal Hilbert Functions

Abstract

In this paper we prove the existence of minimal level artinian graded algebras having socle degree r and type t and describe their h-vector in terms of the r-binomial expansion of t. We also investigate the graded Betti numbers of such algebras and completely describe their extremal resolutions. We also show that any set of points in ? ⁿ whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.     

Convex Pencils of Real Quadratic Forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ?P ⁿ (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere S ⁿ ; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)??2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular?X. In the nondegenerate case we also prove the bound on each specific Betti number b _k (X)??2(k+2).     

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay??s Theorem (Macaulay in Proc. Lond Math Soc 26:531?C555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of P ^r-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne??s Theorem (Hartshorne in Math. IHES 29:5?C48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.     

Betti Numbers of parabolic U(2,1)-Higgs bundles moduli spaces

Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic bundle.      

Topological properties of Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin??s Hilbert schemes of an almost-complex four-manifold X. We compute in this setting their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on X, which are shown to be canonical in K?Ctheory.     

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.     

Growth of Betti numbers

We discuss growth rates of Betti numbers in a family of coverings of a compact cell complex X, when the corresponding L² Betti number of X is zero. We show that the Betti numbers are bounded by a function, sub-linear in the order of the covering. If the appropriate Novikov-Shubin invariant of X is positive, the rate bounds are improved. For well behaved families (such as congruence covers of symmetric spaces), if the L² spectrum of X? has a gap at zero then the growth rate is bounded by the order of the covering raised to a power less than one.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

On the reducibility of the postulation Hilbert scheme

We give a condition in terms of the possible graded Betti numbers compatible with a given Hilbert functionH of 0-dimensional subschemes of ℙ ⁿ which implies the reducibility of the postulation Hilbert scheme and of its subscheme which parametrizes reduced subschemes with Hilbert functionH.     

Resolutions of ideals of any six fat points in

The graded Betti numbers of the minimal free resolution (and also therefore the Hilbert function) of the ideal of a fat point subscheme Z of P² are determined whenever Z is supported at any 6 or fewer distinct points. All results hold over an algebraically closed field k of arbitrary characteristic.     

Is Every Toric Variety an M-Variety?

A complex algebraic variety X defined over the real numbers is called an M-variety if the sum of its Betti numbers (for homology with closed supports and coefficients in ) coincides with the corresponding sum for the real part of X. It has been known for a long time that any nonsingular complete toric variety is an M-variety. In this paper we consider whether this remains true for toric varieties that are singular or not complete, and we give a positive answer when the dimension of X is less than or equal to 3 or when X is complete with isolated singularities.An erratum to this article can be found at     

Hilbert functions and Betti numbers in a flat family

Summary This paper is dedicated to the study of Hilbert functions and Betti numbers of the projective varieties in a flat family. We prove that the Hilbert function H(X _y ,n),y Y-a parameter scheme-is lower semicontinuous for any fixed n. In case Y is integral and noetherian we obtain the well-known fact that the set V Y where H(X _y ,n)is maximal for all n's is open and nonempty. We show also that b_i(X _y )-the i- th Betti number of X_y—is upper semicontinuous for y V. The paper contains also a number of results concerning the relations among the various Betti numbers.Member of G.N.S.A.G.A.-C.N.R. Supported in part by M.P.I. (Italian Minstry of Education).     

Betti numbers of random manifolds

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R ³. We also prove results about higher moments of Betti numbers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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 , 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.     

Lifting monomial ideals

We show how to lift any monomial ideal J in n variables to a saturated ideal J of the same codimension in n -+ t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohornology of l is described. Making general choices for our lifting, we show that l is the ideal of a reduced union of linear varieties with singularities that are "as small as possible" given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for l, and in the case of points we obtain certain k-configurations which we can describe in a very precise way.