Syzygies of projective bundles

In this paper we study defining equations and syzygies among them of projective bundles. We prove that for a given p≥0, if a vector bundle on a smooth complex projective variety is sufficiently ample, then the embedding given by the tautological line bundle satisfies property N_p.     

Separators of points in a multiprojective space

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of and via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a separator to shifts in the minimal multigraded free resolution of the ideal of points.     

Saturation of Jacobian ideals: Some applications to nearly free curves,line arrangements and rational cuspidal plane curves

In this note we describe the minimal resolution of the ideal $I_f$, the saturation of the Jacobian ideal of a nearly free plane curve $C:f=0$. In particular, it follows that this ideal $I_f$ can be generated by at most 4 polynomials. Related general results by Hassanzadeh and Simis on the saturation of codimension 2 ideals are discussed in detail. Some applications to rational cuspidal plane curves and to line arrangements are also given.     

Stability and unobstructedness of syzygy bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E_d1,…,dn on P^N defined as the kernel of a general epimorphism

     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Some equivalent conditions for Huneke’s two conjectures

Abstract

In this paper we give some equivalent conditions for the C. Huneke’s two conjectures concerning the finiteness properties of the local cohomology module , where R is a regular local ring, I is an ideal of R and i ≥ 0 is an integer.     

Arithmetic families of smooth surfaces and equisingularity of embedded schemes

This article deals with the foundations of a theory of equisingularity for families of zero-dimensional sheaves of ideals on smooth algebraic surfaces, in the arithmetic context, i.e., where one works with schemes defined over Dedekind rings. Here, different equisingularity conditions are analyzed and compared, based on one of the following requirements: 1) each member of the the family has the same desingularization tree, 2) the family admits a simultaneous desingularization, 3) a naturally associated family of curves is equisingular. Similar conditions had been investigated, in the context of Complex Local Analytic Geometry, by J. J. Risler. Received: 17 November 1997 / Revised version: 19 April 1999     

On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley-Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley-Reisner and affine monoid algebras. We consider (non-pure) shellable fan’s and the Cohen-Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.     

Complete intersection dimensions and Foxby classes

Let R be a local ring and M a finitely generated R-module. The complete intersection dimension of M-defined by Avramov, Gasharov and Peeva, and denoted -is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities .Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ°φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class A_C(R) for each semidualizing R-complex C.     

Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P³ defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph with each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.     

Some geometric results arising from the Borel fixed property

In this paper, we will give some geometric results using generic initial ideals for the degree reverse lex order. The main goal of the paper is to improve on results of Bigatti, Geramita and Migliore concerning geometric consequences of maximal growth of the Hilbert function of the Artinian reduction of a set of points. When the points have the Uniform Position Property, the consequences are even more striking. Here we weaken the growth condition, assuming only that the values of the Hilbert function of the Artinian reduction are equal in two consecutive degrees, and that the first of these degrees is greater than the second reduction number of the points. We continue to get nice geometric consequences even from this weaker assumption. However, we have surprising examples to show that imposing the Uniform Position Property on the points does not give the striking consequences that one might expect. This leads to a better understanding of the Hilbert function, and the ideal itself, of a set of points with the Uniform Position Property, which is an important open question. In the last section we describe the role played by the Weak Lefschetz Property (WLP) in this theory, and we show that the general hyperplane section of a smooth curve may not have the WLP.     

The projective dimension of three cubics is at most 5

Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension $\mathrm{pd}(R/I)$ of $R/I$ is at most 36, although the example with largest projective dimension he constructed has $\mathrm{pd}(R/I)=5$. Based on computational evidence, it had been conjectured that $\mathrm{pd}(R/I)\le 5$. In the present paper we prove this conjectured sharp bound.     

The geometry of border bases

In this paper we continue the study of the border basis scheme we started in Kreuzer and Robbiano (2008) [16]. The main topic is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal component.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

Hilbert–Kunz multiplicity, McKay correspondence and good ideals¶in two-dimensional rational singularities

Hilbert–Kunz multiplicity is known to be a very mysterious invariant of a ring or an ideal. We will show a very beautiful formula on Hilbert–Kunz multiplicity for integrally closed ideals in two-dimensional Gorenstein rational singularities. In the proof, “McKay correspondence” and “Riemann–Roch formula” play essential roles. Also this formula gives a new significance to “good ideals”. Received: 25 October 2000     

Graded Betti numbers of some embedded rationaln-folds

Supported, in part, by the Natural Sciences and Engineering Research Council of Canada