A Module-theoretic Characterization of Algebraic Hypersurfaces

Miranda-Neto, Cleto B.

Canadian mathematical bulletin, Tome 61 (2018) no. 1, pp. 166-173

Résumé

In this note we prove the following surprising characterization: if $X\,\subset \,{{\mathbb{A}}^{n}}$ is an (embedded, non-empty, proper) algebraic variety deûned over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ of logarithmic vector fields of $X$ is a reflexive ${{O}_{{{\mathbb{A}}^{n}}}}$ -module. As a consequence of this result, we derive that if ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ is a free ${{O}_{{{\mathbb{A}}^{n}}}}$ -module, which is shown to be equivalent to the freeness of the $t$ -th exterior power of ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ for some (in fact, any) $t\,\le \,n$ , then necessarily $X$ is a Saito free divisor.

Détail

DOI : 10.4153/CMB-2016-099-6

Mots-clés : 14J70, 13N15, 32S22, 13C05, 13C10, 14N20, 14C20, 32M25, hypersurface, logarithmic vector field, logarithmic derivation, free divisor