Ramblings on the freeness of affine hypersurfaces

TL;DR

This paper explores the concept of freeness in complex affine hypersurfaces, relating it to projective hypersurfaces, singularity properties, and algebraic invariants, providing new insights into their structural and singularity characteristics.

Contribution

It introduces two types of freeness for affine hypersurfaces and establishes their relations with projective hypersurface freeness, including a proof linking freeness to singularity properties.

Findings

Affine hypersurfaces are free iff all singularities are free in Saito's sense.

Smooth affine hypersurfaces and affine plane curves are always free.

Relations between Jacobian syzygies of affine and projective hypersurfaces.

Abstract

In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, then we relate first the Jacobian syzygies of $D$ and those of $X$. Then we introduce two types of freeness for an affine hypersurface $X$, and prove various relations between them and the freeness of the projective hypersurface $D$. We write down a proof of the folklore result saying that an affine hypersurface is free if and only if all of its singularities are free, in the sense of K. Saito's definition in the local setting. In particular, smooth affine hypersurfaces and affine plane curves are always free. Some other results, involving global Tjurina numbers and minimal degrees of non trivial syzygies are also explored.