On the Jacobian syzygies for generic toric models

TL;DR

This paper investigates the algebraic structure of Jacobian syzygies associated with generic hypersurfaces in affine tori, linking them to arrangements in projective space and computing their minimal graded resolutions.

Contribution

It introduces a novel approach to analyze Jacobian syzygies for generic toric hypersurfaces by relating them to hyperplane arrangements and explicitly computing their minimal graded resolutions.

Findings

Derived minimal graded resolutions of Jacobian algebras.

Established connections between toric hypersurfaces and hyperplane arrangements.

Provided explicit algebraic descriptions for generic cases.

Abstract

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the minimal graded resolutions of the corresponding Jacobian algebra.