## On the size of maximal caps in $$Q^+(5,q)$$.(English)Zbl 1043.51008

The paper of Metsch provides an interesting contribution to the study of caps in finite quadrics.
A cap $${\mathcal C}$$ of the hyperbolic quadric $$Q^+(5, q)$$ is a set of points of $$Q^+(5, q)$$ that does not contain three collinear points. It is called maximal if it is not contained in a larger cap.
For $$q$$ odd it is easily seen that $$| {\mathcal C}| \leq q^3+q^2+q+1$$. For $$q$$ odd D. G. Glynn [Geom. Dedicata 26, No. 3, 273–280 (1988; Zbl 0645.51012)] constructed caps of size $$q^3+q^2+q+1$$ and L. Storme [J. Comb. Theory, Ser. A 87, No. 2, 357–378 (1999; Zbl 0947.51009)] characterized caps of size $$q^3+q^2+q+1$$ as intersections of $$Q^+(5, q)$$ with another quadric, if $$q \geq 3139$$.
Metsch deals with the problem of the maximal cardinality of caps of $$Q^+(5, q)$$, $$q$$ odd, with $$| {\mathcal Q}| < q^3+q^2+q+1$$. His main result is as follows:
Theorem. Let $${\mathcal C}$$ be a maximal cap of $$Q^+(5, q)$$, $$q$$ odd, with $$| {\mathcal Q}| < q^3+q^2+q+1$$. If $$q \geq 4364$$, then $$| {\mathcal Q}| \leq q^3+q^2+2$$.
Note that Storme constructed a maximal cap of size $$q^3+q^2+1$$. It is not clear, whether maximal caps of size $$q^3+q^2+2$$ do exist. Metsch obtained the following information about such an hypothetical cap:
Theorem. Suppose that there exists a maximal cap $${\mathcal C}$$ of $$Q = Q^+(5, q)$$, $$q$$ odd, of size $$q^3+q^2+2$$. If $$q \geq 4364$$, then one of the following cases occurs:
(i) There exists a line $$h$$ of $$Q$$ with $$| h \cap {\mathcal C}| = 2$$. The two planes of $$Q$$ through $$h$$ meet $${\mathcal C}$$ in the two points of $$h \cap {\mathcal C}$$. Every plane of $$Q$$ intersecting $$h$$ in a point $$p \notin {\mathcal C}$$ intersects $${\mathcal C}$$ in a $$q$$-arc that can be extended to a conic by adjoining $$p$$. All other planes of $$Q$$ intersect $${\mathcal C}$$ in a conic.
(ii) There exists a point $$p \in {\mathcal C}$$ such that all planes of $$Q$$ through $$p$$ meet $${\mathcal C}$$ in two points whereas all other planes of $$Q$$ meet $${\mathcal C}$$ in a conic.

### MSC:

 5.1e+22 Blocking sets, ovals, $$k$$-arcs 5.1e+21 Combinatorial structures in finite projective spaces 5.1e+13 Generalized quadrangles and generalized polygons in finite geometry