Zero-dimensional Schemes Research Articles

Differential theory of zero-dimensional schemes

To study a 0-dimensional scheme X in Pn over a perfect field K, we use the module of Kähler differentials ΩR/K1 of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kähler differentials ΩR/Km. One of our main results is a characterization of weakly curvilinear schemes X by the Hilbert polynomials of the modules ΩR/Km which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of X. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules ΩR/Km for a fat point scheme X which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring R into its truncated integral closure R˜. Then we use the corresponding map from the module of Kähler differentials ΩR/K1 to ΩR˜/K1 to find a formula for the Hilbert polynomial HP(ΩR/K1) and a sharp bound for the regularity index ri(ΩR/K1). Next we extend this to formulas for the Hilbert polynomials HP(ΩR/Km) and bounds for the regularity indices of the higher modules of Kähler differentials. As a further application, we characterize uniformity conditions on X using the Hilbert functions of the Kähler differential modules of X and its subschemes.

Two-torsion subgroups of some modular Jacobians

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus [Formula: see text], [Formula: see text] or [Formula: see text]. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the [Formula: see text]-torsion subgroup. Using [Formula: see text]-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton–Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the [Formula: see text]-torsion points. We demonstrate the practicality of our method by computing the [Formula: see text]-torsion of the modular Jacobians [Formula: see text] for [Formula: see text]. As a result of this we are able to verify the generalized Ogg conjecture for these values.

Minimal Terracini Loci in a Plane and Their Generalizations

We study properties of the minimal Terracini loci, i.e., families of certain zero-dimensional schemes, in a projective plane. Among the new results here are: a maximality theorem and the existence of arbitrarily large gaps or non-gaps for the integers x for which the minimal Terracini locus in degree d is non-empty. We study similar theorems for the critical schemes of the minimal Terracini sets. This part is framed in a more general framework.

Variation of stability for moduli spaces of unordered points in the plane

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.

Finite 0-dimensional multiprojective schemes and their ideals

In this paper, we study finite [Formula: see text]-dimensional schemes in product of multiprojective spaces and their ideals. In particular, we describe the set of generators of the ideal defining a [Formula: see text]-dimensional scheme in the case [Formula: see text] and in the case [Formula: see text] with [Formula: see text] for all [Formula: see text]. We also check very ampleness for zero-dimensional schemes of general points in the multiprojective spaces.

Derived categories of coherent sheaves on some zero-dimensional schemes

Let R N = k [ y 1 , … , y N ] / ( y 1 , … , y N ) 2 be the truncated polynomial algebra . Geometrically, it is the algebra of functions on the second infinitesimal neighborhood of a closed point in N -dimensional affine space . In this note we study D b ( R N − mod ) , the bounded derived category of finitely generated R N -modules. We show that for N ≥ 2 the lattice of triangulated subcategories in D b ( R N − mod ) has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. Our homological methods produce some applications to universal localizations of free associative algebras . These applications are based on a relation between triangulated subcategories in D b ( R N − mod ) and universal localizations of a free graded associative algebra in N variables.

Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.

A constructive approach to one-dimensional Gorenstein ${\mathbf {k}}$-algebras

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed.

Computing subschemes of the border basis scheme

A good way of parameterizing zero-dimensional schemes in an affine space [Formula: see text] has been developed in the last 20 years using border basis schemes. Given a multiplicity [Formula: see text], they provide an open covering of the Hilbert scheme [Formula: see text] and can be described by easily computable quadratic equations. A natural question arises on how to determine loci which are contained in border basis schemes and whose rational points represent zero-dimensional [Formula: see text]-algebras sharing a given property. The main focus of this paper is on giving effective answers to this general problem. The properties considered here are the locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley–Bacharach, and strict Cayley–Bacharach properties. The key characteristic of our approach is that we describe these loci by exhibiting explicit algorithms to compute their defining ideals. All results are illustrated by nontrivial, concrete examples.

Vanishing Hessian, wild forms and their border VSP

Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree dge 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers underline{{mathrm {VSP}}} of these forms in the corresponding multigraded Hilbert schemes.

Injective linear series of algebraic curves on quadrics

We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

DEPENDENT SUBSETS OF EMBEDDED PROJECTIVE VARIETIES

Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\rho (X)''\ge \lceil (r+2)/2\rceil$ and that $\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve.

An Application of Liaison Theory to Zero-dimensional Schemes

Given a $0$-dimensional scheme $\mathbb{X}$ in an $n$-dimensional projective space $\mathbb{P}^n_K$ over an arbitrary field $K$, we use liaison theory to characterize the Cayley-Bacharach property of $\mathbb{X}$. Our result extends the result for sets of $K$-rational points given in [8]. In addition, we examine and bound the Hilbert function and regularity index of the Dedekind different of $\mathbb{X}$ when $\mathbb{X}$ has the Cayley-Bacharach property.

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function H of a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2 of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that H is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.

Smoothable zero dimensional schemes and special projections of algebraic varieties

AbstractWe study the degrees of generators of the ideal of a projected Veronese variety to depending on the center of projection. This is related to the geometry of zero dimensional schemes of length 8 in , Cremona transforms of , and the geometry of Tonoli Calabi‐Yau threefolds of degree 17 in .

Lifting zero-dimensional schemes and divided powers

We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.

The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables

We give the stratification by the symmetric tensor rank of all degree d ≥ 9 homogeneous polynomials with border rank 5 and which depend essentially on at least five variables, extending previous works (A. Bernardi, A. Gimigliano, M. Ida, E. Ballico) on lower border ranks. For the polynomials which depend on at least five variables, only five ranks are possible: 5, d + 3, 2d + 1, 3d − 1, 4d − 3, but each of the ranks 3d − 1 and 2d + 1 is achieved in two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme A associated with the polynomial. The polynomial f depends essentially on at least five variables if and only if A is linearly independent (in all cases, f essentially depends on exactly five variables). The polynomial has rank 4d − 3 (resp. 3d − 1, resp. 2d + 1, resp. d + 3, resp. 5) if A has 1 (resp. 2, resp. 3, resp. 4, resp. 5) connected component. The assumption d ≥ 9 guarantees that each polynomial has a uniquely determined associated scheme A. In each case, we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme A.

ON THE SEGRE UPPER BOUND OF THE REGULARITY FOR NON-FAT POINT SCHEMES IN PROJECTIVE SPACES

We study the generalized Segre bound in projective space (mainly in the plane) with respect to zero-dimensional schemes which are more general that the fat point schemes.

Minimal Free Resolutions of Zero-dimensional Schemes in ℙ1 × ℙ1

Let X be a zero-dimensional scheme in ℙ1 × ℙ1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

Hilbert functions and set of points in $${\varvec{\mathbb {P}}^{1}\times {\mathbb {P}}^{1}}$$ P 1 × P 1

In this paper we study the problem of classifying the Hilbert functions of zero-dimensional schemes in \(\mathbb P^1\times \mathbb P^1\). In particular, in the main result of the paper we give conditions to determine some Hilbert functions of set of points in \(\mathbb P^1\times \mathbb P^1\) and we describe geometrically these schemes. Moreover, we show that the Hilbert functions of these schemes depend only on the distribution of the points on a set of \((1,0)\) and \((0,1)\)-lines.