Vanishing products of one-forms and critical points of master functions

TL;DR

This paper investigates the structure of critical points of master functions associated with hyperplane arrangements, revealing their relation to resonance varieties and providing conditions for their nonemptiness and structure.

Contribution

It establishes bounds on the codimension of critical loci, links them to resonance varieties, and characterizes when certain subspaces of differential forms correspond to critical points.

Findings

Critical loci have codimension at most p under certain conditions.

Critical loci are unions of intersections of level sets of rational master functions.

Conditions are provided for the nonemptiness and precise codimension of critical loci.

Abstract

Let \A be an affine hyperplane arrangement in $\C^\ell$ with complement $U$. Let $f_1, \..., f_n$ be linear polynomials defining the hyperplanes of \A, and $A^\cdot$ the algebra of differential forms generated by the 1-forms $d \log f_1, \..., d \log f_n$. To each $l \in \C^n$ we associate the master function $\Phi=\Phi_l = \prod_{i=1}^n f_i^{l_i}$ on $U$ and the closed logarithmic 1-form $\omega= d \log \Phi$. We assume $\omega$ is an element of a rational linear subspace $D$ of $A^1$ of dimension $q>1$ such that the multiplication map $\bigwedge^k(D) \to A^k$ is zero for $p<k\leq q$. With this assumption, we prove every component of the critical locus $\crit(\Phi)$ of $\Phi$ has codimension at most $p$, and $\crit(\Phi)$ is a union of intersections of level sets of rational master functions. We give conditions that guarantee $\crit(\Phi)$ is nonempty and every component has codimension equal to $p$, in terms of syzygies among polynomial master functions. If \A is $p$-generic, then $D$ is contained in the degree $p$ resonance variety $\R^p(\A)$ -- in this sense the present work complements previous work on resonance and critical loci of master functions. Any arrangement is 1-generic; in case $p=1$ we give a precise description of $\crit(\Phi_l)$ in case $l$ lies in an isotropic subspace $D$ of $A^1$, using the multinet structure on \A corresponding to $D\subseteq \R^1(\A)$. This is carried out in detail for the Hessian arrangement. Finally, for arbitrary $p$ and \A, we establish necessary and sufficient conditions for a set of integral one-forms to span such a subspace, in terms of nested sets of \A, using tropical implicitization.