## 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:

 5.1e+22 Blocking sets, ovals, $$k$$-arcs

### Keywords:

complete arcs; derivation plane

### Citations:

Zbl 0452.51007; Zbl 0589.51020; Zbl 0605.51008; Zbl 1008.51018
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