On finite k-nets in the complex projective plane
Igor Vladimirovich Dolgachev, Janis Stipins · Deep Blue (University of Michigan) · 2007
A finite (k, d)-net in the complex projective plane is an arrangement of k · d lines with the following property: The lines may be partitioned into k sets A 1,..., A k of d-lines each, in such a way that any two of the d-gons ∪ A i and ∪ A j are perspective from every line in every other A m. Equivalently, the d-gons are k completely reducible members of a pencil of degree d curves with distinct base points. We prove that there are no (4, d)-nets in the complex projective plane with d > 3. (There is a well-known and projectively unique example of a (4, 3)-net in the complex projective plane.) Via the equivalence described above, our result implies that for d > 3, a pencil of degree d curves with distinct base points has at most three completely reducible members.