On (k,n)-arcs and the falsity of the Lunelli-Sce conjecture

Summary

INTRODUCTION

A (k,n)-arc in PG(2,q) is a set of k points, such that some n, but no n+1, of them are collinear. The maximum value of k for which a (k,n)-arc exists in PG(2,q) will be denoted by m (n)_2,q - Clearly m(l)_2,q = 1 and m(q+l)_2,q = q² + q + 1, and so from now on we assume that 2 ≤ n ≤ q.

The following two theorems are well-known, and proofs may be found in Chapter 12 of Hirschfeld [12].

THEOREM 1.1:

1. (i) m(n)_2,q ≤ (n-l)q + n.

2. (ii) (Cossu [8]). For n≤q, equality can occur in (i) only if n | q.

A ((n-l)q + n, n)-arc is known as a maximal arc.

THEOREM 1.2: There exists a maximal arc in PG(2,q) when

(i) n = q, for any q, a maximal arc being the complement of a line,

(ii) (Denniston [10]). q = 2^h, and n is any divisor of q.

It is not known whether maximal arcs exist when q is odd and 2 ≤ n < q, although it has been proved by Thas [16] that there are no (2q+3, 3)- arcs in PG(2, 3^h) for h > 1.