On finite <italic>k</italic>-nets in the complex projective plane.

Abstract

A finite (<italic>k, d</italic>)-net in the complex projective plane is an arrangement of <italic>k &middot; d</italic> lines with the following property: The lines may be partitioned into <italic>k</italic> sets A ₁,..., A <italic>_k</italic> of <italic>d</italic>-lines each, in such a way that any two of the <italic>d</italic>-gons &cup; A <italic>_i</italic> and &cup; A <italic>_j</italic> are perspective from every line in every other A <italic>_m</italic>. Equivalently, the <italic>d</italic>-gons are <italic>k</italic> completely reducible members of a pencil of degree <italic> d</italic> curves with distinct base points. We prove that there are no (4, <italic>d</italic>)-nets in the complex projective plane with <italic>d</italic> > 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 <italic>d</italic> > 3, a pencil of degree <italic>d</italic> curves with distinct base points has at most three completely reducible members.