For q = p^r with a prime p ≥ 7 such that q ≡ 1 or 19 (mod 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 Γ isomorphic to A_6 of P G(2, q), we investigate the geometric properties of the (unique) Γ-orbit O of size 90 such that the 1-point stabilizer of Γ in its action on O is a cyclic group of order 4. Here O lies either in P G(2, q) or in P G(2, q^2) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q different from 421, then O is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601, 661. Interestingly, O is the smallest known complete arc in P G(2, 601) and in P G(2,661). Computations are carried out by MAGMA.
Transitive A_6-invariant k-arcs in PG(2, q)
GIULIETTI, Massimo;MARCUGINI, Stefano;PAMBIANCO, Fernanda
2013
Abstract
For q = p^r with a prime p ≥ 7 such that q ≡ 1 or 19 (mod 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 Γ isomorphic to A_6 of P G(2, q), we investigate the geometric properties of the (unique) Γ-orbit O of size 90 such that the 1-point stabilizer of Γ in its action on O is a cyclic group of order 4. Here O lies either in P G(2, q) or in P G(2, q^2) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q different from 421, then O is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601, 661. Interestingly, O is the smallest known complete arc in P G(2, 601) and in P G(2,661). Computations are carried out by MAGMA.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
