相关论文: On the Dedekind different of a Cayley-Bacharach sc…

Given a 0-dimensional scheme $\mathbb{X}$ in the projective $n$-space $\mathbb{P}^n_K$ over a field $K$, we are interested in studying the K\"ahler different of $\mathbb{X}$ and its applications. Using the K\"ahler different, we…

Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given…

The Cayley-Bacharach property, 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…

We extend the minimal model theorem to the 3-dimensional schemes which are projective and have semistable reduction over the spectrum of a Dedekind ring.

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X…

We characterise Dedekind rings among not necessarily Noetherian domains by a property of their module homomorphisms. Our proof relies on a homological algebra argument.

We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular…

Given an ACM set $\mathbb{X}$ of points in a multiprojective space $\mathbb{P}^m\times\mathbb{P}^n$ over a field of characteristic zero, we are interested in studying the K\"ahler different and the Cayley-Bacharach property for…

Let X be a separated finite type scheme over a noetherian base ring K. There is a complex C(X) of topological O_X-modules on X, called the complete Hochschild chain complex of X. To any O_X-module M - not necessarily quasi-coherent - we…

Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$.…

Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a field $K$, we study the K\"ahler differential algebra $\Omega_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules…

For a 0-dimensional scheme $\mathbb{X}$ in $\mathbb{P}^n$ over a perfect field $K$, we first embed the homogeneous coordinate ring $R$ into its truncated integral closure $\widetilde{R}$. Then we use the corresponding map from the module of…

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of To\"en and Vaqui\'{e}. In this paper we show that when $X$ is…

A good way of parametrizing 0-dimensional schemes in an affine space $\mathbb{A}_K^n$ has been developed in the last 20 years using border basis schemes. Given a multiplicity $\mu$, they provide an open covering of the Hilbert scheme ${\rm…

We prove that for Noetherian, smooth, separated, integral, finite type schemes $X$ and $Y$ over an excellent Dedekind domain $R$, that are properly birational over $R$, we have $R^if_{*}\mathcal{O}_X \cong R^ig_{*} \mathcal{O}_Y$ and $R^i…

We describe the canonical module of a simplicial affine semigroup ring $\mathbb{K}[S]$ and its trace ideal. As a consequence, we characterize when $\mathbb{K}[S]$ is nearly Gorenstein in terms of arithmetic properties of the semigroup $S$.…

In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp.…

We describe some recent work concerning Gorenstein liaison of codimension two subschemes of a projective variety. Applications make use of the algebraic theory of maximal Cohen-Macaulay modules, which we review in an Appendix.

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i. e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over…

In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the projective plane. In this paper, we generalize Hansen's results…