\(p\)-adic ovoids (Q1919316)

A subset \({\mathbf O}\) of a 3-dimensional projective space \(\Pi= ({\mathcal P}, {\mathcal L})\) is called an ovoid if each line \(l\in {\mathcal L}\) intersects \({\mathbf O}\) in at most two points, and for each point \(p\in {\mathbf O}\), the union of all lines \(l\) with \({\mathbf O} \cap l= \{p\}\) is a plane; the so-called tangent plane of \({\mathbf O}\) in \(p\). The analogous object in a projective plane \(\pi\) is called an oval. The author studies the projective space \(\Pi= \text{PG} (3,K)\) defined over a locally compact field with a non-archimedean valuation, i.e. \(K\) is either a finite extension of a \(p\)-field \(\mathbb{Q}_p\) or the field of Laurent series over a finite field. The topology of \(K\) induces topologies on the space of points, lines and planes of \(\text{PG} (3,K)\), respectively. A closed ovoid \({\mathbf O} \subseteq \text{PG} (3,K)\) is defined to be a differentiable ovoid if for all sequences \(q_n,r_n\) in \({\mathbf O}\) convergent to a point \(p\in {\mathbf O}\) with \(q_n \neq r_n\) such that the sequence \(q_n \vee r_n\) converges to a line \(l\), this line \(l\) is contained in the tangent plane of \({\mathbf O}\) at \(p\). The definition of a differentiable oval in \(\text{PG}(2,K)\) is analogous. The aim of the paper under review is a characterization of the differentiable closed ovoids in \(\text{PG} (3,K)\) by their homomorphic images in \(\text{PG} (3,k)\), where \(k\) is the (finite) residue field of \(K\). The following theorem is proved: Let \({\mathbf O} \subseteq \text{PG} (3,K)\) be a closed subset, not contained in a plane. We require that for each homomorphism \(\mu: \text{PG} (3,K)\to \text{PG} (3,k)\), the image \(\mu ({\mathbf O})\) is one of the following: 1. An ovoid in \(\text{PG} (3,k)\) or 2. a cone in \(\text{PG} (3,k)\) or 3. the union of two planes of \(\text{PG} (3,k)\) minus some points on their common line, or 4. a subset of a plane in \(\text{PG} (3,k)\). Then, \({\mathbf O}\) is either a differentiable ovoid or a differentiable cone or the union of two planes. Conversely, each differentiable ovoid in \(\text{PG} (3,K)\) satisfies this condition. There is an analogous result, proved by the same author, for ovals in the projective plane \(\text{PG} (2,K)\). The main strategy of the proof of the current theorem consists of a reduction to this planar result. The key theorem used for this reduction states that a closed subset \({\mathbf O}\) of \(\text{PG} (3,K)\) with at least two points is a differentiable ovoid if and only if for each plane \(e\), the intersection \({\mathbf O} \cap e\) is either empty, a point or a differentiable oval in \(e\).