Abstract view
A moduletheoretic characterization of algebraic hypersurfaces


Published:20170208
Printed: Mar 2018
Cleto B. MirandaNeto,
Departamento de Matemática, Universidade Federal da Paraíba, 58051900 João Pessoa, PB, Brazil
Abstract
In this note we prove the following surprising characterization:
if
$X\subset {\mathbb A}^n$ is an (embedded, nonempty, 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.
MSC Classifications: 
14J70, 13N15, 32S22, 13C05, 13C10, 14N20, 14C20, 32M25 show english descriptions
Hypersurfaces Derivations Relations with arrangements of hyperplanes [See also 52C35] Structure, classification theorems Projective and free modules and ideals [See also 19A13] Configurations and arrangements of linear subspaces Divisors, linear systems, invertible sheaves Complex vector fields
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
