location:  Publications → journals → CMB
Abstract view

# A module-theoretic characterization of algebraic hypersurfaces

Published:2017-02-08
Printed: Mar 2018
• Cleto B. Miranda-Neto,
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil
 Format: LaTeX MathJax PDF

## Abstract

In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor.
 Keywords: hypersurface, logarithmic vector field, logarithmic derivation, free divisor
 MSC Classifications: 14J70 - Hypersurfaces 13N15 - Derivations 32S22 - Relations with arrangements of hyperplanes [See also 52C35] 13C05 - Structure, classification theorems 13C10 - Projective and free modules and ideals [See also 19A13] 14N20 - Configurations and arrangements of linear subspaces 14C20 - Divisors, linear systems, invertible sheaves 32M25 - Complex vector fields

 top of page | contact us | privacy | site map |