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A module-theoretic characterization of algebraic hypersurfaces

  • Cleto Brasileiro Miranda-Neto,
    Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil
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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 hypersurface, logarithmic vector field, logarithmic derivation, 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
 

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