We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if, roughly speaking, the cohomology groups ${{H}^{m}}\left( M,\,\mathcal{L} \right)$ can be computed directly from the cohomology algebra ${{H}^{*}}\left( M,\,\mathbb{C} \right)$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety ${{\mathcal{V}}_{1}}\left( M \right)$ and the Resonance variety ${{\mathcal{R}}_{1}}\left( M \right)$.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$, having the points of multiplicity $\ge \,3$ situated on two lines in $\mathcal{A}$, say ${{H}_{0}}$ and ${{H}_{\infty }}$. Then we show that the non-local irreducible components of the first resonance variety ${{\mathcal{R}}_{1}}(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $P$ in ${{\mathbb{C}}^{2}}={{\mathbb{P}}^{2}}\text{ }\backslash \text{ }{{H}_{\infty }}$ whose sides are in $\mathcal{A}$ and for which ${{H}_{0}}$ is a diagonal.

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a ‘multinet’, a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.