相关论文: The K\"ahler different of a 0-dimensional scheme

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…

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…

Given a 0-dimensional scheme $\mathbb{X}$ in a projective space $\mathbb{P}^n_K$ over a field $K$, we characterize the Cayley-Bacharach property of $\mathbb{X}$ in terms of the algebraic structure of the Dedekind different of its…

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…

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…

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…

The scalars in vector multiplets of N=2 supersymmetric theories in 4 dimensions exhibit `special Kaehler geometry', related to duality symmetries, due to their coupling to the vectors. In the literature there is some confusion on the…

In this paper, we study the geometry of points in complex projective space that satisfy the Cayley-Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez…

Let $K$ be a nonarchimedean local field of characteristic zero with valuation ring $R$, for instance, $K=\mathbb{Q}_p$ and $R=\mathbb{Z}_p$. We prove a general integral geometric formula for $K$-analytic groups and homogeneous $K$-analytic…

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of…

Let $X$ be an Abelian threefold. We prove a formula, conjectured by the first author, expressing the Euler characteristic of the generalized Kummer schemes $K^nX$ of $X$ in terms of the number of plane partitions. This computes the…

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study the space of generalized K\"ahler (GK) structures defined by a symplectic form $F\in \alpha$…

In every point of a K\"ahler manifold there exist special holomorphic coordinates well adapted to the underlying geometry. Comparing these K\"ahler normal coordinates with the Riemannian normal coordinates defined via the exponential map we…

The authors construct the global Macaulay inverse system for a zero-dimensional subscheme Z of projective n-space P^n, from the local inverse systems of the irreducible components of Z. They show that when Z is locally Gorenstein a generic…

This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized K\"ahler geometry from sigma models with additional spinorial superfields.…

Two-dimensional (2,2) supersymmetric nonlinear sigma models can be described in (2,2), (2,1) or (1,1) superspaces. Each description emphasizes different aspects of generalized K\"ahler geometry. We investigate the reduction from (2,2) to…

Strong K\"ahler with Torsion is the target space geometry of $(2,1)$ and $(2,0)$ supersymmetric nonlinear sigma models. We discuss how it can be represented in terms of Generalised Complex Geometry in analogy to the Gualtieri map from the…

Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail,…

We compute the Euler characteristics of the generalized Kummer schemes associated to $A\times Y$, where $A$ is an abelian variety and $Y$ is a smooth quasi-projective variety. When $Y$ is a point, our results prove a formula conjectured by…