The complete classification of (3, 3)-nets and of (3, 4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of pencil-type line arrangements. For arrangements consisting of  14 lines and having points of multiplicity  5, we show that the non-triviality of the monodromy on the first cohomology H 1 (F) of the associated Milnor fiber F implies the arrangement is of reduced pencil-type. In particular, the monodromy is determined by the combinatorics in such cases.

Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.

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