Enumeration of surfaces containing an elliptic quartic curve

Abstract

A very general surface of degree at least four in P 3 contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces in 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.so F 4 contains the pencil of elliptic quartics A 1 − tQ 2 , A 2 + tQ 1 , t ∈ P 1 ; setting t = ∞, we find Q 1 , Q 2 . Similarly, we get Q 1 − tA 2 , Q 2 + tA 1 . This is one and the same pencil. But there is also A 1 − tA 2 , Q 2 + tQ 1 . In general, these 2 pencils are disjoint. Looking at them as curves in X = G(2, F 2 ), we actually get a Plückerembedded conic, (A 1 −tA 2 )∧(Q 2 +tQ 1 ) = A 1 ∧Q 2 +t(A 1 ∧Q 1 −A 2 ∧Q 2 )−t 2 A 2 ∧Q 1 , disjoint from Y (see (3.1)). In particular, capping each conic against the Plücker hyperplane class Π = −c 1 A, we find 2. As before, we may write deg NL(W, d) = H 33 ∩ NL (W, d).

Enumeration of hypersurfaces with prescribed non-isolated singular subschemes

“…Avritzer & Vainsencher [34], [3] obtained an explicit description of the component W eqc of elliptic quartics of the Hilbert scheme Hilb 4t (P 3 ). This has been used in [9] for enumerating curves in cer-tain Calabi-Yau 3-folds, and in [6] for studying Noether-Lefschetz loci of systems of surfaces in P 3 . G. Gotzmann [13] has shown that Hilb 4t (P 3 ) consists of two irreducible components; the second one parameterizes unions of a plane quartic curve and a zero dimensional subcheme of P 3 of length 2.…”

“…Pulling back E d , D d in (7) to W, we may as well simplify notation and assume W = W smooth. We now argue as in [6] and [33]. Recall D d is a direct image of a sheaf over W × P n (cf.…”

Foliations singular along a curve

“…See a script for the production of random examples in [21]. Just as in [8], the proof of Theorem A uses Grothendieck-Riemann-Roch [13, 15.2, p. 286] to show the polynomial nature and bound the degree of q W (d).…”

Hypersurfaces in P^5 containing unexpected subvarieties

“…Asking surfaces of a given degree to contain say, a (few) line(s), or a conic, a twisted cubic, etc., defines subvarieties, the so called Noether-Lefschetz loci, in the appropriate projective space. There are polynomial formulas for their degrees, [4], [14].…”

“…An argument employing Grothendieck-Riemann-Roch (cf. [4]) shows that there is a formula for the degree of W d ⊂ PF d as a polynomial in d. Actually we got by interpolation the following polynomial of degree 54 (cf. [16] for a script):…”