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A Module-theoretic Characterization of Algebraic Hypersurfaces
Published online by Cambridge University Press: 20 November 2018
Abstract
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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.
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