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

Let \A be an affine hyperplane arrangement in ${\text{C}}^{\mathrm{\ell }}$ with complement $U$. Let $f_1,\dot{\mathrm{.}}\mathrm{.},f_n$ be linear polynomials defining the hyperplanes of \A, and $A^{\cdot }$ the algebra of differential forms generated by the 1-forms $d \log f_1, .…, d \log f_n$\mathrm{.}Toeach$l \in \C^n$weassociatethemasterfunction$\Phi=\Phi_l = \prod_{i=1}^n f_i^{l_i}$on$U$ and the closed logarithmic 1-form $\omega =d\log \mathrm{\Phi }$. 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 $\text{crit}(\mathrm{\Phi })$ of $\mathrm{\Phi }$ has codimension at most $p$, and $\text{crit}(\mathrm{\Phi })$ is a union of intersections of level sets of rational master functions. We give conditions that guarantee $\text{crit}(\mathrm{\Phi })$ 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 ${\mathbb{R}}^p(\text{A})$ – 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 $\text{crit}({\mathrm{\Phi }}_l)$ in case $l$ lies in an isotropic subspace $D$ of $A^1$, using the multinet structure on \A corresponding to $D\subseteq {\mathbb{R}}^1(\text{A})$. 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.