Ovoidal linear spaces (Q1394812)

A finite linear space \(L = (P, {\mathcal L})\) of dimension \(d\) not less than 3 is called \(n\)-ovoidal, if there is an \(n\)-cap (a subset of \(n\) points intersecting each line in at most two points) satisfying the following conditions: (i) the number of external lines is smaller than or equal to the number of secant lines; (ii) for any two different points \(x\) and \(y\) in the \(n\)-cap, there is a unique tangent line through \(x\) and \(y\) intersecting any other tangent through \(x\) and any other tangent through \(y\); (iii) three distinct tangent sets intersect in a point: (iv) a secant line and a tangent set intersect in at most one point. In the paper, the main result is the following theorem: If \(L\) is an \(n\)-ovoidal linear space, then either \(n=2\) and \(l\) is a double near-pencil or \(n = q^2n+ 1\), \(L\) is a Galois space \(\text{PG}(3,q)\) and the \(n\)-cap is an ovoid.