Counting triangles in smooth cubic hypersurfaces

Abstract

We propose and study the notion of triangles in smooth cubic hypersurfaces. We prove that for a generic cubic n-fold X (\(n\ge 2\)), the variety of triangles in X is of dimension \(3n-6\). We show that on a generic cubic n-fold, the triangles with a given edge can be parametrized by an open subset of a quintic hypersurface in \(\mathbb {P}^{n-1}\). In the case of a generic cubic threefold, we show that the locus of the opposite vertices for triangles with a given edge form a curve of degree 10. As a corollary, we get an interesting enumerative result on the number of triangles satisfying some restrictions.