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

Abstract

Let \A be an affine hyperplane arrangement in $\C^\ell$ with complement $U$. Let $f_1,\dot{.}.,f_n$ be linear polynomials defining the hyperplanes of \A, and $A^{\cdot }$ the algebra of differential forms generated by the 1-forms $d\log f_1,\dot{.}.,d\log f_n$. 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 $p<k\le q$. With this assumption, we prove every component of the critical locus $\crit(Φ)$ of $\Phi$ has codimension at most $p$, and $\crit(Φ)$ is a union of intersections of level sets of rational master functions. We give conditions that guarantee $\crit(Φ)$ 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 $\crit(Φ_l)$ 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.

v2: Major revision. Example 3.17 added to illustrate effects of singularities, thanks to an anonymous referee. To appear in "Arrangements of Hyperplanes - Sapporo 2009," Adv. Studies in Pure Math., in press. v3: final version; minor corrections