Codimension one decompositions and Chow varieties

E. Carlini

Home/Archive/arXiv

arXiv · math/0410602

Codimension one decompositions and Chow varieties

E. Carlini

View original source ↗Read source document ↗Export metadata ↓

Abstract

A presentation of a degree $d$ form in $n+1$ variables as the sum of homogenous elements ``essentially'' involving $n$ variables is called a {\em codimension one decomposition}. Codimension one decompositions are introduced and the related Waring Problem is stated and solved. Natural schemes describing the codimension one decompositions of a generic form are defined. Dimension and degree formulae for these schemes are derived when the number of summands is the minimal one; in the zero dimensional case the scheme is showed to be reduced. These results are obtained by studying the Chow variety $Δ_{n,s}$ of zero dimensional degree $s$ cycles in $\PP^n$. In particular, an explicit formula for $\degΔ_{n,s}$ is determined.

Explore related subjects

math.AG math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

E. Carlini. 2004-10-28. Codimension one decompositions and Chow varieties. https://arxiv.org/abs/math/0410602

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Good minimal models with nowhere vanishing holomorphic $1$-forms

Popa and Schnell show that any holomorphic 1-form on a smooth projective variety of general type has zeros. In this article, we show that a smooth good minimal model has a holomorphic 1-form without zero if and only if it admits an analytic fiber bundle structure over a positive dimensional abelian variety.

math.AG↗

arXiv2026

The automorphism groups of Kummer surfaces in characteristic two and their complex analogues

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the complex $K3$ surfaces with the same Picard lattice as these Kummer surfaces. The paper has two appendices.

math.AG↗

arXiv2026

Algebraic Zhou valuations

In this paper, we generalize Zhou valuations, originally defined on complex domains, to the framework of general schemes. We demonstrate that an algebraic version of the Jonsson--Mustaţă conjecture is equivalent to the statement that every Zhou valuation is quasi-monomial. By introducing a mixed version of jumping numbers and Tian functions associated with valuations, we obtain characterizations of a valuation being a Zhou valuation or computing some jumping number using the Tian functions. Furthermore, we establish the correspondence between Zhou valuations in algebraic settings and their counterparts in analytic settings.

math.AG↗