Enumerative formulae for ruled cubic surfaces and rational quintic curves
type: Item
instance of: scholarly article
title: Enumerative formulae for ruled cubic surfaces and rational quintic curves
zbMATH Open document ID: 0602.14049
DOI: 10.1007/BF02621930
author: Daniel Coray
author: I. Vainsencher
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140066
review text: 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.
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 3970934
zbMATH Keywords: ruled cubic surfaces
zbMATH Keywords: rational quintic curves
zbMATH Keywords: Grassmannian
reviewed by: Robert Speiser
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3970934