Enumeration of surfaces containing an elliptic quartic curve

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.


Introduction
The Noether-Lefschetz theorem asserts that all curves contained in a very general surface F of degree at least four in P 3 are complete intersections. This is usually rephrased by saying that the Picard group is Z. Noether-Lefschetz theory shows that, roughly speaking, each additional generator for PicF decreases the dimension of the locus of such F in P N = |O P 3 (d)|, d ≥ 4.
Let W be a closed, irreducible subvariety of the Hilbert scheme of curves in P 3 with Hilbert polynomial p W (t). Let us denote by N L(W, d) the subset of P N defined by the requirement that the surface contains some member of W .
The purpose of this note is to address the question of determining the degree of N L(W, d) for the family of elliptic quartic curves in P 3 .
When W is the family of lines, or conics, or twisted cubics, formulas for N L(W, d) have been found in [9]. There as here, we follow the strategy of using Bott's formula as explained in [5]. We get a polynomial formula (4.3) valid for d ≥ 5. We also compute the degree (38475) of the locus of quartic surfaces containing an elliptic quartic curve. The case of quartic surfaces is not covered by the formula essentially because the map that forgets the curve shrinks dimensions: generically, it contracts a pair of disjoint pencils, see 4.2.

there is a polynomial formula
Let W be a closed, irreducible subvariety of the Hilbert scheme of curves in P 3 with Hilbert polynomial p W (t). Let (2.1) W × P N p2 denote the projection maps from W × P N . Castelnuovo-Mumford regularity [12] shows that for all d >> 0, the subset N L(W, d) of pairs (C, F ) in W × P N such that the curve C is contained in the surface F is a projective bundle over W via p 1 . We have For instance, if W is the Grassmannian of lines in P 3 , then p W (d) = d + 1 and so dim N L(W, d) = N − (d − 3), d ≥ 1. Let us denote by N L(W, d) the subset of P N defined by the requirement that the surface contains some member of W . In other words, with notation as in (2.1), N L(W, d) = p 2 ( N L(W, d)).
We assume henceforth that the general member of W is a smooth curve.
Proof. Let C ⊂ W × P 3 be the universal curve. Likewise, let F ⊂ P N × P 3 be the universal surface of degree d. Write C, F for their pullbacks to W × P N × P 3 . We have the diagram of sheaves over Y : By construction, the slant arrow ρ vanishes at a point (C, F, x) ∈ W × P N × P 3 if and only if x ∈ F ∩ C. We have C ⊂ F when the previous condition holds for all x ∈ C (point with values in any C-algebra). Thus N L(W, d) is equal to the scheme of zeros of ρ along the fibers of the projection p 12 : C → W × P N . Recalling [1, (2.1),p. 14], this is the same as the zeros of the adjoint section of the direct image vector bundle p 12⋆ (O C ( F )). Let By Castelnuovo-Mumford and base change theory, there is an integer d 0 such that E d is a vector bundle of rank p W (d) for all d ≥ d 0 (=regularity, see Remarks 2.2.). In fact, it fits into the exact sequence of vector bundles over W , (trivial vector bundle with fiber) the space of polynomials of degree d. Taking the projectivization, and pulling back to P N × W , we get By construction, ρ vanishes precisely over N L (W, d). This shows that we actually get Since the rank of E d and codimension of N L(W, d) agree, it follows that N L(W, d) represents the top Chern class of O P N (1) ⊗ E d (cf. [6, 3.2.16, p. 61]). This is the key to the calculation of degrees below. The map is generically injective by Noether-Lefschetz theory [8], cf. Corollary 4.2 below. Therefore the degree of N L(W, d) can be computed upstairs. Namely, setting where τ = rankE d . Expanding the top Chern class and pushing forward to W , we arrive at (2.6) deg with w = dim W . Since E d is the pushforward of a sheaf on W × P 3 , we may apply Grothendieck-Riemann-Roch [6, p. 286] to express the Chern character of E d as Notice that the right hand side is a polynomial in d of degree ≤ 3. Since c w is a polynomial of degree w on the coefficients of the Chern character, we deduce that c w (E d ) is a polynomial in d of degree ≤ 3w.

2.2.
Remarks. (1) The assertion that E d , as defined in (2.2), is a vector bundle of rank p W (d) holds for all d beyond d 0 = the maximal Castelnuovo-Mumford regularity of the members of W . For instance, if W is the family of lines in P 3 , then d 0 = 1.
(2) For the case of elliptic quartic curves presented below, we note that the regularity of the ideals x 2 1 , x 2 2 and x 1 x 2 , x 2 1 , x 3 2 is 3, whereas for x 2 0 , x 0 x 1 , x 0 x 2 2 , x 4 1 it is 4. The last two ideals are representatives of the closed orbits in W . An argument of semi-continuity shows that d 0 = 4 works for all members of W , see [2]. Nevertheless, the map in (2.5) is generically injective only for d ≥ 5, cf. (4.2) below. Notice that the full Hilb 4t P 3 has a "ghost" component with regularity index ≥ 5, see [7].

elliptic quartics
We consider now the case of surfaces of degree ≥ 4 containing an elliptic quartic curve in P 3 . Thus, a general member C 4 of W is the intersection of two quadric surfaces. The parameter space W is described in [2] and has been used in [5] to enumerate curves in Calabi-Yau 3-folds. For the convenience of the reader, we summarize below its main features.
The Noether-Lefschetz locus of quartic surfaces containing some C 4 is slightly exceptional. This is a case when the map (2.5) fails to be generically injective (cf. Corollary 4.2): it actually shrinks dim N L(W, d) = 34 to dim N L(W, d) = 33. Indeed, if a quartic surface F contains some general elliptic quartic C 4 , then F must contain the two pencils 3.1. Next we give an outline of the calculation. Put the Grassmannian of pencils of quadrics in P 3 .
The diagram below summarizes the construction of W as explained in [2].
bundle of degree 2 divisors on the varying ℓ ⊂ p; X = the blow-up of X along Y and X = the blow-up of X along Z.
be the tautological subbundle of rank 2 over our Grassmannian of pencils of quadrics. There is a natural map of vector bundles over X induced by multiplication, with generic rank 8. It drops rank precisely over Z. It induces a rational map κ : X G(8, F 3 ). Blowing up X along Z, we find the closure X ⊂ G(8, F 3 ) × X of the graph of κ. Similarly, up on X we have a subbundle B ⊂ F 3 × X of rank 8 and a multiplication map with generic rank 19. The scheme of zeros of 19 µ 4 is equal to Y . Indeed, it can be checked that each fiber of B is a linear system of cubics which • either has base locus equal to a curve with Hilbert polynomial p W (t) = 4t • or is of the form p · F ⋆ ⋆ 2 , meaning the linear system with fixed component a plane p, and F ⋆ ⋆ 2 denoting an 8-dimensional space of quadrics which define a subscheme of p of dimension 0 and degree 2.
The exceptional divisor E is a P 8 -bundle over Y . The fiber of E over (p, y 1 + y 2 ) ∈ Y is the system of quartic curves in the plane p which are singular at the doublet y 1 + y 2 . Precisely, if x 0 , . . . , x 3 denote homogeneous coordinates on P 3 , assuming p := x 0 , ℓ = x 0 , x 1 , a typical doublet has homogeneous ideal of the form Given a non-zero quartic g in this ideal, we may form the ideal J = x 2 0 , x 0 x 1 , x 0 f, g . It can be checked that J contains precisely 19 independent quartics and its Hilbert polynomial is correct. In fact, any such ideal is 4-regular (in the sense of Castelnuovo-Mumford). Moreover, up on X we get a subbundle C ⊂ F 4 × X of rank 19. Each of its fibers over X is a system of quartics that cut out a curve with the correct Hilbert polynomial. The multiplication map Corollary 4.2 (ii) below. Hence the degree of the image N L(W, d) ⊂ P N is given by c 16 E d , cf. (2.6).
The above description suffices to feed in Bott's localization formula with all required data. Indeed, X inherits a C ⋆ -action, with (a lot) of isolated fixed points. The vector bundle D d → X is equivariant; ditto for E d . Bott's formula reads [4], (3.2) c The equivariant classes on the r.h.s are calculated in two steps. Below we set d = 5 for simplicity.
First, find the C ⋆ -weight decomposition of the fibers of the vector bundles E d and τ at each fixed point. Then, since the fixed points are isolated, the equivariant Chern classes c T i are just the symmetric functions of the weights.
For instance, for the tangent bundle τ, say at the fixed point corresponding to the pencil p = Each of the 16 fractions, xixj x k x l , on the right hand side symbolizes a 1-dimensional subspace of τ p where C ⋆ acts with character t xi+xj −x k −x l . The denominator in (3.2) is the corresponding equivariant top Chern class, to wit ( . The numerator in (3.2) requires finding the monomials of degree 5 that survive modulo the ideal We are left with 20 (=rank E 5 ) terms, . (20 factors.) In practice, all these calculations are made substituting x i for suitable numerical values, cf. the computer algebra scripts in [13].
3.2. the case d = 4. Presently p 2 : N L(W, 4) → N L(W, 4) is no longer generically injective. It shrinks dimension by one: a general fiber is a disjoint union of P 1 's (cf. Corollary 4.2 (iii) below). Explicitly, say Similarly, 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 Plücker-embedded conic, (3.1)). In particular, capping each conic against the Plücker hyperplane class Π = −c 1 A, we find 2. As before, we may write The latter integral can be computed via Bott's formula and we get 38475, cf. the script in [13]. This has been found independently in [3] with different techniques, using [10].

The fibers of p 2
The main result needed to validate the above enumeration is the following.

4.1.
Proposition. Let C ⊂ P 3 be a smooth irreducible curve of degree e and genus g. Let d ≫ 0 and let F ⊂ P 3 be a general surface of degree d containing C. Then C is the only effective divisor of degree e and arithmetic genus g on F .
Proof. By [8,Cor.II.3.8] we have that Pic(F ) is freely generated by its hyperplane section H and C. Let C ′ be an effective divisor of degree e and arithmetic genus g on F . Then there are two integers a, b such that, on F , we have C ′ ∼ aH + bC. Now e = H · C ′ = ad + be, so that By adjunction formula we have Note that e 2 − dC 2 > 0 by (4.2), whence b = ±1. If b = −1 we have from (4.1) a = 2e d ∈ Z, as d ≫ 0. Therefore we deduce that b = 1 and a = 0, that is C ′ ∼ C. Since C 2 < 0 we must have C ′ = C.

4.2.
Corollary. Let W be an irreducible subvariety of a Hilbert scheme component of curves in P 3 of degree e and arithmetic genus g with general member smooth. Let P N = P(H 0 (O P 3 (d))). Then (i) there is a d 0 such that for all d ≥ d 0 the projection map p 2 : N L(W, d) → P N is generically injective.
(ii) If W is the family of elliptic quartics then we can take d 0 = 5, i.e., p 2 is generically one-toone for d ≥ 5 and (iii) for d = 4, the general fiber of N L(W, 4) p2 −→ N L(W, 4) is two disjoint P 1 's.
Proof. We know from (2.4) that p 1 : N L(W, d) → W is a projective bundle. Hence N L(W, d) = p 2 ( N L(W, d)) is irreducible and a general element F ∈ N L(W, d) can be identified with a general hypersurface of degree d containing a general C ∈ W . Hence assertion (i) follows from the propostion. Assertion (ii) also follows, except for d = 8. In this case, with the notation as in (4.1), if b = −1 we would get a = 1, whence C ′ ∼ H − C so that C would be contained in a plane, which is absurd. If b = 1, may proceed as at the end of the proof of the above proposition.
Moreover there is no curve D on F such that D ∼ C and D ∼ 2H − C for then 2C ∼ 2H giving the contradiction 0 = C 2 = H 2 = 4. This proves that, in this case, the general fiber of p 2 is two disjoint P 1 's, 4.3. The formula. In view of Proposition 2.1, it suffices to find the degrees of N L(W, d) for 3·16+1 values of d ≥ 5 and interpolate. This is done in [13]. We obtain (2 27 · 3 9 · 5 2 · 7 2 · 11 · 13).

4.4.
Acknowledgments. The authors thank the referee for kindly pointing out to them a sizable portion of the argument that required clarification. They are also grateful to Kristian Ranestad and Rahul Pandharipande for correcting an error in §3.2 and for calling their attention to [3] and [10].