Caps on Hermitian varieties and maximal curves

Abstract

A lower bound for the size of a complete cap of the polar space H(n, q 2 ) associated to the non-degenerate Hermitian variety U n is given; this turns out to be sharp for even q when n = 3. Also, a family of caps of H(n, q 2 ) is constructed from F q 2 -maximal curves. Such caps are complete for q even, but not necessarily for q odd.

Bounds on partial ovoids and spreads in classical generalized quadrangles

“…Unfortunately not much is known about large maximal partial spreads of Q(4, q), and the largest known such partial spread is still much smaller than the theoretical bound q 2 − q + 1, q odd. Finally it is interesting to note that in [9] Hirschfeld and Korchmáros construct maximal partial spreads of Q(5, q) (in fact they construct maximal partial ovoids of H(3, q 2 )) from GF(q 2 )-maximal curves for even q. However, their examples have size q 2 + 1 + 2gq, where g is the genus of the curve used, which is also still much larger then our obtained lower bound (except for the trivial case of rational algebraic curves which have genus 0 and yield maximal partial ovoids of size q 2 + 1).…”

Commuting polarities and maximal partial ovoids of H(4,q2)

“…Hirschfeld and Korchmáros [10], proved that a maximal partial ovoid of H (n, q 2 ) has at least q 2 +1 points. They showed that this lower bound is sharp for n = 3 and even q. Aguglia, Ebert and Luyckx [1] dealt with small partial ovoids in H (3, q 2 ).…”