Conference abstracts

Session B2 - Graph Theory and Combinatorics

July 15, 15:30 ~ 16:20 - Room B7

Planar arcs

Simeon Ball

Universitat Polit├Ęcnica de Catalunya, Spain   -

Let $\mathrm{PG}_{2}({\mathbb F}_q)$ denote the projective plane over ${\mathbb F}_q$. An arc (or planar arc) of $\mathrm{PG}_{2}({\mathbb F}_q)$ is a set of points in which any $3$ points span the whole plane. An arc is complete if it cannot be extended to a larger arc.

In 1967 Beniamino Segre proved that the set of tangents to a planar arc of size $q+2-t$, when viewed as a set of points in the dual plane, is contained in an algebraic curve of small degree $d$. Specifically, if $q$ is even then $d=t$ and if $q$ is odd then $d=2t$.

The main result to be presented here is the following theorem.

Theorem 1: Let $S$ be a planar arc of size $q+2-t$ not contained in a conic. If $q$ is odd then $S$ is contained in the intersection of two curves, sharing no common component, each of degree at most $t+p^{\lfloor \log_p t \rfloor}$.

This leads directly to the following theorem.

Theorem 2: If $q$ is odd then an arc of size at least $q-\sqrt{q}+3+\max \{ \sqrt{q}/p,\frac{1}{2} \}$ is contained in a conic.

There are examples of complete arcs of size $q+1-\sqrt{q}$ in $\mathrm{PG}_2({\mathbb F}_q)$ when $q$ is square, first discovered by Kestenband in 1981. These arcs are the intersection of two curves of degree $t$.It has long been conjectured that if $q \neq 9$ is an odd square then any larger arc is contained in a conic.

Joint work with Michel Lavrauw (Sabanci University).

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