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

Abstract

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