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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.

##### MSC:
 5.1e+22 Blocking sets, ovals, $$k$$-arcs 5.1e+21 Combinatorial structures in finite projective spaces
##### Keywords:
complete caps; hermitian varieties
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