Cayley-Bacharach Property Research Articles

The Kähler different of a 0-dimensional scheme

Given a 0-dimensional scheme [Formula: see text] in the projective [Formula: see text]-space [Formula: see text] over a field [Formula: see text], we are interested in studying the Kähler different of [Formula: see text] and its applications. Using the Kähler different, we characterize the generic position and Cayley–Bacharach properties of [Formula: see text] in several certain cases. When [Formula: see text] is in generic position, we prove a generalized version of the Apéry–Gorenstein–Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the Kähler different and the Cayley–Bacharach property.

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson conjecture

Let S⊂P3 be a very general sextic surface over complex numbers. Let M(H,c2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c2. In this article we study the configuration of certain locally complete intersection zero-dimensional subschemes on S satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general member of each component of the moduli space for small c2. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of M(H,11) and prove the conjecture partially. We will also show that M(H,c2) is irreducible for c2≤10.

Kähler Differentials for Fat Point Schemes in ℙ1×ℙ1

Let 𝕏 be a set of K-rational points in ℙ1×ℙ1 over a field K of characteristic zero, let 𝕐 be a fat point scheme supported at 𝕏, and let R𝕐 be the bihomogeneous coordinate ring of 𝕐. In this paper we investigate the module of Kahler differentials ΩR𝕐∕K1. We describe this bigraded R𝕐-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support 𝕏 is a complete intersection or an almost complete intersection in ℙ1×ℙ1. Moreover, we introduce a Kahler different for 𝕐 and use it to characterize ACM reduced schemes in ℙ1×ℙ1 having the Cayley–Bacharach property.

Identifiability for a Class of Symmetric Tensors

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley–Bacharach property, to prove that, when the Kruskal’s ranks of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e., the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal’s and the reshaped Kruskal’s criteria.

The Dedekind different of a Cayley–Bacharach scheme

Given a 0-dimensional scheme [Formula: see text] in a projective space [Formula: see text] over a field [Formula: see text], we characterize the Cayley–Bacharach property (CBP) of [Formula: see text] in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley–Bacharach schemes by Dedekind’s formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

On the Cayley-Bacharach Property

The Cayley-Bacharach Property (CBP), which has been classically stated as a property of a finite set of points in an affine or projective space, is extended to arbitrary 0-dimensional affine algebras over arbitrary base fields. We present characterizations and explicit algorithms for checking the CBP directly, via the canonical module, and in combination with the property of being a locally Gorenstein ring. Moreover, we characterize strict Gorenstein rings by the CBP and the symmetry of their affine Hilbert function, as well as by the strict CBP and the last difference of their affine Hilbert function.

On the Cayley-Bacharach property and the construction of vector bundles

We study the Cayley-Bacharach property on complex projective smooth varieties of dimension n ⩾ 2 for zero dimensional subscheme defined by the zero set of the wedge of r − n + 1 global sections of a rank r ⩾ n vector bundle, and give a construction of high rank reflexive sheaves and vector bundles from codimension 2 subschemes.

Codes associated to the zero-schemes of sections of vector bundles

We consider the algebraic geometric codes associated to the zero-schemes of sections of vector bundles on a smooth projective variety. We give lower bounds for the minimum distances of the codes exploiting the Cayley–Bacharach property of zero-dimensional subschemes.

Level Algebras, Lex Segments, and Minimal Hilbert Functions

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 ℙ n whose Hilbert function has first difference as described above, must satisfy the Cayley-Bacharach property.

N-TYPE VECTORS AND THE CAYLEY-BACHARACH PROPERTY

ABSTRACT We develop an “arithmetic” of -type vectors in order to discuss the relationship between -type vectors and Hilbert functions. We then use this arithmetic to determine for which Hilbert functions it is true that every set of points with that Hilbert function has the Cayley-Bacharach Property. As a second application, we determine the degree of every point in a -configuration.