k-nets embedded in a projective plane over a field

TL;DR

This paper studies the structure of $k$-nets in projective planes over fields, proving non-existence for $k extgreater=5$ in large positive characteristic and providing a new proof in characteristic zero.

Contribution

It extends the non-existence results of $k$-nets to large positive characteristic fields and introduces a novel proof method applicable in characteristic zero.

Findings

No embedded $k$-nets for $k extgreater=5$ in large positive characteristic

New proof of non-existence in characteristic zero

Provides bounds on the size of $k$-nets in different characteristics

Abstract

We investigate $k$-nets with $k\geq 4$ embedded in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$; they are line configurations in $PG(2,\mathbb{K})$ consisting of $k$ pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a $k$-net is the same, the order of the $k$-net. If $\mathbb{K}$ has zero characteristic, no embedded $k$-net for $k\geq 5$ exists; see [1,2]. Here we prove that this holds true in positive characteristic $p$ as long as $p$ is sufficiently large compared with the order of the $k$-net. Our approach, different from that used in [1,2], also provides a new proof in characteristic zero. [1] J. Stipins, Old and new examples of k-nets in P2, math.AG/0701046. [2] S. Yuzvinsky, A new bound on the number of special fibers in a pencil of curves, Proc. Amer. Math. Soc. 137 (2009), 1641-1648.