  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.
