Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

TL;DR

This paper introduces four new families of homogeneous free divisors, studies their Jacobian ideals' blowup algebras, and explores conditions for maximal analytic spread and homaloidal divisors, advancing understanding in algebraic geometry.

Contribution

It presents four new families of free divisors and analyzes their blowup algebras, establishing Cohen-Macaulay properties and characterizing maximal analytic spread.

Findings

Rees algebra and special fiber are Cohen-Macaulay for all families.

Characterization of maximal analytic spread using cohomology and asymptotic depth.

Provides a homological criterion for homaloidal divisors.

Abstract

Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are birational.