Combinatorial cubic surfaces and reconstruction theorems (Q2918493)

The main object of the paper under review is a cubic hypersurface \(V\subset \mathbb P_K^N\). It is assumed to be generically reduced, which means that the cubic \(K\)-form defining \(V\) does not acquire a multiple factor after extending scalars to \(\overline K\). The main problem under consideration is the following: reconstruct the field \(K\) and the subscheme \(V\subset \mathbb P_K^N\) from the set of \(K\)-points \(V(K)\) endowed with some additional combinatorial data of geometric origin. In the case of surfaces (which is in the focus of author's attention), such data include the set of smooth \(K\)-points of \(V\) together with ``collinearity'' relation and a set of subsets called ``plane sections''. The problem consists in finding combinatorial constraints on these data which guarantee that they actually come from some cubic surface and producing a combinatorial reconstruction procedure. The author proceeds along these lines, providing a framework for such a procedure under mild genericity assumptions.NEWLINENEWLINEIn the appendix, the author explains in some detail his motivation. It is related to questions of Mordell--Weil type: can one get all \(K\)-points of \(V\) from a finite set of \(K\)-points by drawing secants and tangents? Since the strategy based on heights, as in the case of cubic curves, may break down (as shown by numeric experiments which are also discussed in the appendix), the suggested reconstruction technique (which, in a sense, imitates the Hrushovski--Zilber model-theoretic approach to reconstructing projective plane) may be a reasonable alternative.NEWLINENEWLINEFor the entire collection see [Zbl 1242.11004].