Enumeration of surfaces containing an elliptic quartic curve

Cukierman, F.; Lopez, A.; Vainsencher, I.

doi:10.1090/S0002-9939-2014-11998-8

Enumeration of surfaces containing an elliptic quartic curve
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by F. Cukierman, A. F. Lopez and I. Vainsencher PDF

Proc. Amer. Math. Soc. 142 (2014), 3305-3313 Request permission

Abstract:

A very general surface of degree at least four in $\mathbb {P}^{3}$ contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces in $\mathbb {P}^{3}$ of degree at least five which contain some elliptic quartic curves. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.

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