Old and new examples of k-nets in P^2

Abstract

In this paper, we present a number of examples of k-nets, which are special configurations of lines and points in the projective plane. Such a configuration can be regarded as the union of k completely reducible elements of a pencil of complex plane curves; equivalently it can be regarded as a set of k polygons in the complex projective plane that satisfy a condition of mutual perspectivity and nondegenerate intersection. For each example, we describe its construction, combinatorial properties, and parameter space. Most of the examples are historical, although perhaps not very well-known; our only essentially new example is a 3-net of pentagons which does not realize a group. The existence of this example settles a question posed by S. Yuzvinsky.