Logarithmic vector fields along smooth divisors in projective spaces

TL;DR

This paper investigates the conditions under which a smooth divisor in a projective space can be uniquely reconstructed from its associated sheaf of logarithmic vector fields, revealing a specific class of divisors for which this is possible.

Contribution

It establishes a precise criterion linking the reconstructibility of a smooth divisor to the Sebastiani-Thom type of its defining equation.

Findings

Reconstruction is possible if and only if the divisor's defining equation is of Sebastiani-Thom type.

Provides a characterization of divisors based on their sheaf of logarithmic vector fields.

Connects geometric properties of divisors with algebraic conditions on their defining equations.

Abstract

We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type.