Transitive A_6-invariant k-arcs in PG(2,q)

TL;DR

This paper studies special 90-arcs in projective planes with A_6 symmetry, showing their completeness for large q and identifying minimal examples in certain cases, using computational methods.

Contribution

It demonstrates the existence and completeness of specific A_6-invariant 90-arcs in PG(2,q) for large q, and identifies the smallest known complete arcs in PG(2,601) and PG(2,661).

Findings

90-arcs are complete for q=349, 409, 529, 601, 661

The 90-arc in PG(2,601) and PG(2,661) are the smallest known complete arcs

Existence of a unique conjugacy class of A_6 groups acting on PG(2,q)

Abstract

For $q=p^r$ with a prime $p\ge 7$ such that $q \equiv 1$ or $19\pmod {30},$ the desarguesian projective plane $PG(2,q)$ of order $q$ has a unique conjugacy class of projectivity groups isomorphic to the alternating group $A_6$ of degree 6. For a projectivity group $\Gamma\cong A_6$ of $PG(2,q)$, we investigate the geometric properties of the (unique) $\Gamma$-orbit $\mathcal{O}$ of size 90 such that the 1-point stabilizer of $\Gamma$ in $\mathcal O$ is a cyclic group of order 4. Here $\mathcal O$ lies either in $PG(2,q)$ or in $PG(2,q^2)$ according as 3 is a square or a non-square element in $GF(q)$. We show that if $q\geq 349$ and $q\neq 421$, then $\mathcal O$ is a 90-arc, which turns out to be complete for $q=349, 409, 529, 601,661.$ Interestingly, $\mathcal O$ is the smallest known complete arc in $PG(2,601)$ and in $PG(2,661).$ Computations are carried out by MAGMA.