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A class of complete arcs in multiply derived planes. (English) Zbl 1043.51005

B. C. Kestenband [Geom. Dedicata 11, 107–117 (1981; Zbl 0452.51007)] constructed arcs of size \(q^2-q+1\) in \(PG(2, q^2)\). The completeness of these arcs has been shown by J. C. Fisher, J. W. P. Hirschfeld and J. A. Thas [Ann. Discr. Math. 30, 243–250 (1986; Zbl 0589.51020)] and – independently – by E. Boros and T. Szönyi [Combinatorics 6, 261–268 (1986; Zbl 0605.51008)]. G. Rinaldi [Abh. Math. Semin. Univ. Hamb. 71, 197–203 (2001; Zbl 1008.51018)] showed that a Kestenband arc of \(PG(2, q^2)\) remains an arc in the Hall plane of the same order. If \(q^2 > 9\), then the Kestenband arc is also complete in the Hall plane.
Bonisoli and Rinaldi extend this result from Hall planes which can be seen as planes obtained from \(PG(2, q^2)\) by a single derivation to multiply derived planes. Their main theorem reads as follows:
Theorem. Let \(K\) be a Kestenband arc of \(P = PG(2, q^2)\). If \(q > 3\), then there exists for each \(r < q-1\) an \(r\)-fold derived plane \(P_r\) of \(P\) such that \(K\) is a complete arc of \(P_r\).

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs
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