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Caps on Hermitian varieties and maximal curves. (English) Zbl 1043.51007
Let \(U\) be the non-degenerate Hermitian variety of \(P = PG(n, q^2)\). The maximal subspaces of \(P\) contained in \(U\) are called the generators of \(U\). A cap of \(U\) is a set \({\mathcal C}\) of points of \(U\) such that each generator has at most one point in common with \(U\). If each generator has exactly one point in common with \({\mathcal C}\), then \({\mathcal C}\) is called an ovoid.
J. A. Thas [Geom. Dedicata 10, 135–143 (1981; Zbl 0458.51010)] showed that if \(n\) is even, the non-degenerated Hermitian variety of \(P\) does not contain ovoids. For \(n\) odd it is only known that ovoids exist in the non-degenerated Hermitian variety of \(PG(3, q^2)\) [see J. A. Thas and S. E. Payney, Geom. Dedicata 52, 227–253 (1994; Zbl 0804.51007)].
The authors construct complete caps in \(PG(3, q^2)\), \(q\) even (a cap is called complete, if it is not contained in a larger cap):
Let \(X\) be an algebraic curve of \(PG(n, q^2)\) which is also considered as an algebraic curve of \(PG(n, F)\), where \(F\) is the algebraic closure of \(GF(q^2)\). Let \(g\) be the genus of \(X\), and denote by \(X_q\) the set of points of \(X\) in \(PG(n, q^2)\). By the theorem of Hasse and Weil, the cardinality of \(X_q\) is less than or equal to \(q^2 + 1 + 2gq\). An algebraic curve \(X\) with \(| X_Q| = q^2 + 1 + 2gq\) is called maximal.
Due to a theorem of G. Korchmáros and F. Torres [Compos. Math. 128, No. 1, 95–113 (2001; Zbl 1024.11044)], up to isomorphism the set \(X_q\) is contained in the Hermitian variety \(U\) of \(PG(n, q^2)\). The main result of Hirschfeld and Korchmáros reads as follows:
Theorem. Let \(X\) be a maximal algebraic curve of \(PG(n, q^2)\) embedded in the non-degenerated Hermitian variety \(U\) of \(PG(n, q^2)\).
(a) \(X_q\) is a cap of \(U\) of cardinality \(q^2+1+2q\).
(b) If \(n = 3\) and if \(q\) is even, then \(X_q\) is complete.

51E21 Blocking sets, ovals, \(k\)-arcs
51E20 Combinatorial structures in finite projective spaces
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