Enumerative formulae for ruled cubic surfaces and rational quintic curves
For varieties in \({\mathbb{P}}^ 3\), the classical enumerative geometers obtained their results for curves of degree \(\leq 4\) and surfaces of degree \(\leq 2.\) Here the program is continued with new results about ruled cubic surfaces and rational quintic curves. For ruled cubic surfaces (all are singular, each doubled along a line), there are 4 types, and the authors find the degree of the closure, in the \({\mathbb{P}}^{19}\) of cubic surfaces, of the locus of each type. The authors also find 105 rational quintic curves through 10 general points of \({\mathbb{P}}^ 3\); this is related to the preceding by an elementary construction. For the proofs, one considers the cubic surfaces which are double along a given line; varying the line produces bundles over the Grassmannian, and one computes with these.
scientific article; zbMATH DE number 3970934