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Commentarii Mathematici Helvetici
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Enumerative formulae for ruled cubic surfaces and rational quintic curves

Authors: Vainsencher, Israel; Coray, Daniel F.;

Enumerative formulae for ruled cubic surfaces and rational quintic curves

Abstract

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.

Country
Germany
Keywords

510.mathematics, Grassmannian, Enumerative problems (combinatorial problems) in algebraic geometry, rational quintic curves, ruled cubic surfaces, Article

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Average
Average
Average
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gold