  A central question in arrangement theory is to determine whether the
characteristic polynomial $\Delta_q$ of the algebraic monodromy acting on the
homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a
complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection
lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that
the multiplicities of the factors of $\Delta_1$ corresponding to certain
eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti
numbers $\beta_p(\mathcal{A})$, which in turn are extracted from
$L(\mathcal{A})$. When $\mathcal{A}$ defines an arrangement of projective lines
with only double and triple points, this leads to a combinatorial formula for
the algebraic monodromy. To obtain these results, we relate nets on the
underlying matroid of $\mathcal{A}$ to resonance varieties in positive
characteristic. Using modular invariants of nets, we find a new realizability
obstruction (over $\mathbb{C}$) for matroids, and we estimate the number of
essential components in the first complex resonance variety of $\mathcal{A}$.
Our approach also reveals a rather unexpected connection of modular resonance
with the geometry of ${\rm SL}_2(\mathbb{C})$-representation varieties, which
are governed by the Maurer--Cartan equation.
