Zero-dimensional Schemes Research Articles

POSTULATION OF ZERO-DIMENSIONAL SCHEMES ON A SMOOTH QUADRIC SURFACE

Let Q be a smooth quadric surface and Z ⊂ Q a zero-dimensional scheme. We study the postulation of a general union of Z and prescribed numbers of fat points with multiplicity 2 and 3. AMS Subject Classification: 14N05, 14H99

Interpolation, Bridgeland stability and monomial schemes in the plane

Given a zero-dimensional scheme Z, the higher-rank interpolation problem asks for the classification of slopes μ such that there exists a vector bundle E of slope μ satisfying Hi(E⊗IZ)=0 for all i. In this paper, we solve this problem for all zero-dimensional monomial schemes in P2. As a corollary, we obtain detailed information on the stable base loci of Brill–Noether divisors on the Hilbert scheme of points on P2. We prove the correspondence between walls in the Bridgeland stability manifold and walls in the Mori chamber decomposition of the effective cone conjectured in [2] for monomial schemes. We determine the Harder–Narasimhan filtration of ideal sheaves of monomial schemes for suitable Bridgeland stability conditions and, as a consequence, obtain a new resolution better suited for cohomology computations than other standard resolutions such as the minimal free resolution.

ZERO-DIMENSIONAL SCHEMES IN THE PLANE

Let Z P 2 be a zero-dimensional scheme. Fix t2 N. In this paper we study the following question: nd assumptions on Z and t such that h 1 (IA(t)) < h 1 (IZ(t)) for all A ( Z and check if t does not exist for a certain class of schemes Z.

Finding inverse systems from coordinates

Let I be a homogeneous ideal in R=K[x0,…,xn], such that R/I is an Artinian Gorenstein ring. A famous theorem of Macaulay says that in this instance I is the ideal of polynomial differential operators with constant coefficients that cancel the same homogeneous polynomial F. A major question related to this result is to be able to describe F in terms of the ideal I. In this note we give a partial answer to this question, by analyzing the case when I is the Artinian reduction of the ideal of a reduced (arithmetically) Gorenstein zero-dimensional scheme Γ⊂Pn. We obtain F from the coordinates of the points of Γ.

Postulation of a Union X⊂ℙr, r≥4, of a Given Zero-Dimensional Scheme and Several General Lines

We study the Hilbert function of the union X of a given zero-dimensional scheme Z⊂ℙr, r≥4, and t general lines of ℙr, proving, for example, that if t≫deg(Z), then X has maximal rank.

Multiple points, scheme rank and symmetric tensor rank

In this paper we prove some upper bounds for the symmetric tensor rank of a symmetric tensor (or a homogeneous polynomial) in terms of integers associated to any zero-dimensional scheme evincing the scheme rank of the homogeneous polynomial.

On the geometry of Hermitian one-point codes

Here we describe the algebraic geometry of one-point codes from the Hermitian curve. In particular, we employ zero-dimensional schemes in the plane to characterize the minimum-weight codewords of their dual codes, providing explicit formulas for their number. We discuss also some natural improvements of the duals of Hermitian one-point codes by means of geometric arguments. Finally, some cohomological tools are developed to characterize the small-weight codewords of such codes.

Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Let X⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈 〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg.

On partial polynomial interpolation

The Alexander–Hirschowitz theorem says that a general collection of k double points in P n imposes independent conditions on homogeneous polynomials of degree d with a well known list of exceptions. We generalize this theorem to arbitrary zero-dimensional schemes contained in a general union of double points. We work in the polynomial interpolation setting. In this framework our main result says that the affine space of polynomials of degree ⩽ d in n variables, with assigned values of any number of general linear combinations of first partial derivatives, has the expected dimension if d ≠ 2 with only five exceptional cases. If d = 2 the exceptional cases are fully described.

On the Hilbert function of zero-dimensional schemes in $${{\mathbb P}^1 \times {\mathbb P}^1}$$

Let \({Q = {\mathbb P}^1 \times {\mathbb P}^1}\) and let \({X\subset Q}\) be a zero-dimensional scheme. The results in this paper give the possibility of computing, under certain hypotheses, the Hilbert function of a zero-dimensional scheme in Q that is not ACM. In particular we show how, under some conditions on X, its Hilbert function changes when we add points to X lying on a (1, 0) or (0, 1)-line. As a particular case we show also that if X is ACM this result holds without any additional hypothesis.

A syzygetic approach to the smoothability of zero-dimensional schemes

We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this question for zero-dimensional schemes of regularity two. This invariant imposes obstructions for smoothability in general, and it completely answers the question of smoothability for certain zero-dimensional schemes of low degree. The tools of this paper also lead to other results about Hilbert schemes of points, including a characterization of nonsmoothable zero-dimensional schemes of minimal degree in every embedding dimension d ⩾ 4 .

Naïve noncommutative blowups at zero-dimensional schemes

In an earlier paper [D.S. Keeler, D. Rogalski, J.T. Stafford, Naïve noncommutative blowing up, Duke Math. J. 126 (2005) 491–546, MR 2120116], we defined and investigated the properties of the naïve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naïvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in qgr - A is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper [D. Rogalski, J.T. Stafford, A class of noncommutative projective surfaces, in press] to prove that a large class of noncommutative surfaces can be written as naïve blowups.

Closed subschemes of quadrics with large groups of linear automorphisms

Let Qn ⊂ P n+1 , n ≥ 2, be an n-dimensional smooth quadric hypersurface. Fix P ∈ Qn. Set Gn := Aut(Qn) and Gn(P ): ={f ∈: f (P )= P }. We first study low dimensional linear subspaces of H 0 (P n+1 , OPn+1(k)). Then we study the zero-dimensional schemes Z ⊂ Qn such that f (Z )= Z for all f ∈ Gn(P ) and length(Z) is small.

The Horace Method for Error-Correcting Codes

Here we apply the so-called Horace method for zero-dimensional schemes to error-correcting codes on complete intersections. In particular, we obtain sharper estimates on the minimum distance.

The geometric interpretation of Fröberg–Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space

The study of infinitesimal deformations of a variety embedded in projective space requires, at ground level, that of deformation of a collection of points, as specified by a zero-dimensional scheme. Further, basic problems in infinitesimal interpolation correspond directly to the analysis of such schemes. An optimal Hilbert function of a collection of infinitesimal neighbourhoods of points in projective space is suggested by algebraic conjectures of R. Fröberg and A. Iarrobino. We discuss these conjectures from a geometric point of view. They give, for each such collection, a function (based on dimension, number of points, and order of each neighbourhood) which should serve as an upper bound to its Hilbert function (Weak Conjecture). The Strong Conjecture predicts when the upper bound is sharp, in the case of equal order throughout. In general we refer to the equality of the Hilbert function of a collection of infinitesimal neighbourhoods with that of the corresponding conjectural function as the Strong Hypothesis. We interpret these conjectures and hypotheses as accounting for the infinitesimal neighbourhoods of projective subspaces naturally occurring in the base locus of a linear system with prescribed singularities at fixed points. We develop techniques and insight toward the conjectures' verification and refinement. The main result gives an upper bound on the Hilbert function of a collection of infinitesimal neighbourhoods in Pn based on Hilbert functions of certain such subschemes of Pn−1. Further, equality occurs exactly when the scheme has only the expected linear obstructions to the linear system at hand. It follows that an infinitesimal neighbourhood scheme obeys the Weak Conjecture provided that the schemes identified in codimension one satisfy the Strong Hypothesis. This observation is then applied to show that the Weak Conjecture does hold valid in Pn for n⩽3. The main feature here is that the result is obtained although the Strong Hypothesis is not known to hold generally in P2 and, further, P2 presents special exceptional cases. Consequences of the main result in higher dimension are then examined. We note, then, that the full weight of the Strong Conjecture (and validity of the Strong Hypothesis) are not necessary toward using the main theorem in the next dimension. We end with the observation of how our viewpoint on the Strong Hypothesis pertains to extra algebraic information: namely, on the structure of the minimal free resolution of an ideal generated by linear forms.

Computing zero-dimensional schemes

This paper is a natural continuation of Abbott et al. [Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L., 2000. Computing ideals of points. J. Symbolic Comput. 30, 341–356] further generalizing the Buchberger–Möller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the problems which can be solved by the algorithm: an approach which looks to be readily applicable in several areas. Implementation issues are also addressed, especially for computations over Q where a trace-lifting paradigm is employed. We give a complexity analysis of the new algorithm for fat points in affine space over Q . Tables of timings show the new algorithm to be efficient in practice.

Cayley–Bacharach and evaluation codes on complete intersections

Hansen (Appl. Algebra Eng. Comm. Comput. 14 (2003) 175) uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P 2 . In this paper, we generalize Hansen's results from P 2 to P m ; we also show that the hypotheses of Hansen (2003) may be weakened. The proof is succinct and follows by combining the Cayley–Bacharach Theorem and the bounds on evaluation codes obtained in Hansen (Zero-Dimensional Schemes (Ravello, 1992), de Gruyter, Berlin, 1994, pp. 205–211).

A symbolic test for (i,j)-uniformity in reduced zero-dimensional schemes

Let Z denote a finite collection of reduced points in projective n-space and let I denote the homogeneous ideal of Z. The points in Z are said to be in (i,j)-uniform position if every cardinality i subset of Z imposes the same number of conditions on forms of degree j. The points are in uniform position if they are in (i,j)-uniform position for all values of i and j. We present a symbolic algorithm that, given I, can be used to determine whether the points in Z are in (i,j)-uniform position. In addition it can be used to determine whether the points in Z are in uniform position, in linearly general position and in general position. The algorithm uses the Chow form of various d-uple embeddings of Z and derivatives of these forms. The existence of the algorithm provides an answer to a question of Kreuzer.

Graded duality and generalized fractions

A global algebraic description of dualizing and residual complexes for projective varieties in terms of generalized fractions is given and applied to the study of zero-dimensional schemes.

Inverse System of a Symbolic Power II. The Waring Problem for Forms

The classical Waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of sdth powers of linear forms. We show that its solution is implied by a result of J. Alexander and A. Hirschowitz, concerning the Hilbert functions of the ideal of functions Vanishing to order two at a generic set of s points in Pr − 1. Using Macaulay′s inverse systems, we show that the Alexander-Hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are dth powers of a given set of general linear forms. We also determine the dimension of the family of degree d forms that have additive decompositions of length s. We then study several notions of length for forms f, having to do with the kind of length-s, zero-dimensional schemes Z in Pr − 1 whose defining ideal I(Z) annihilates the inverse system of f. When Z is to consist of distinct points, we obtain the above length of additive decomposition of f. When Z is smoothable we obtain the "smoothable length" of f; when Z is arbitrary, we obtain a "scheme length" of f. All these lengths are at least as large as the dimension of the vector space of all order-i partial derivates of f, for each i. The above-mentioned length functions are distinct. Using results about the existence of nonsmoothable Gorenstein point singularities in codimension 4, we show that when r = 5 there are forms f of scheme length s, which are not in the closure of the family of forms having additive decompositions of length s. Finally, we propose a new set of Waring problems for forms, using these lengths.