Appendix

J.-L. Colliot-Thélène Orsay

Throughout this appendix the ground field is algebraically closed of characteristic zero.

Proposition 1 Let X ⊂ ℙ^{3} be a smooth projective surface. If the number of reduced and irreducible curves C of degree d lying on X is not finite, then there exists such a curve C ⊂ X whose self-intersection (C.C) is nonnegative.

For the proof we let G denote the open set of the Hilbert scheme of curves in ℙ^{3} corresponding to integral (i.e. reduced and irreducible) curves of degree d. This is a scheme of finite type over the ground field. Let W ⊂ G × ℙ^{3} be the reduced closed subset whose points are pairs (c, x) where c is a point with associated curve C and x ∈ ℙ^{3} lies on C. For each c ∈ G, the fibre of W → G above c is just the projective integral curve C.

Let Z ⊂ G × X be the trace of W on G × X. The projection map Z → G is projective. Its image is a closed subset F ⊂ G. A curve C with associated point c lies on X if and only if c belongs to F, and the inverse image of such a c in Z is precisely the curve C ⊂ X. If each component of F is of dimension zero, then F is finite, and there are only finitely many curves C of degree d lying on X. If that is not the case, then F contains at least one irreducible curve T. Two distinct fibres C_{0} and C_{1} of Z → G above rational points of T define integral curves in the same algebraic family on X. Hence (C_{0}.C_{0}) = (C_{0}.C_{1}) ≥ 0, as claimed.

Proposition 2 Let X ⊂ ℙ^{3} be a smooth projective surface of degree n and let C ⊂ X be a reduced and irreducible curve, possibly singular, of degree d in ℙ^{3}. If n > d + 1, then the intersection number (C.C) of C on the surface X is strictly negative.

The canonical sheaf K on X is O_{X}(n − 4).

The formula for the arithmetic genus of C ⊂ X is well known to be

2p_{a}(C) − 2 = (C.C) + (C.K) = (C.C) + (n − 4)d

(see [Ha; Chapter V, Exercise 1.3, page 366]). Here one should recall that p_{a}(C) is by definition the dimension of H^{1}(C, O_{C}).

Suppose first that C is contained in a plane ℙ^{2} ⊂ ℙ^{3}. Then the formula for the arithmetic genus of C ⊂ ℙ^{2} is

2p_{a}(C) − 2 = d(d − 3).

Thus

(C.C) + (n − 4)d = d(d − 3),

and hence

(C.C) = d(d + 1 − n)

is strictly negative as soon as n > d + 1.

Suppose that C is not contained in a plane, hence in particular d ≥ 3. In classical parlance, such a curve is called nondegenerate. For such curves we have the Castelnuovo bound : If d is even, p_{a}(C) ≤ (d^{2}/4) − d + 1; if d is odd, p_{a}(C) ≤ (d^{2} − 1)/4 − d + 1.

Comparing with the formula for p_{a}(C), we find that (C.C) < 0 if n > d/2+2, whether d is even or odd. This completes the proof of the proposition.

Let us comment on the Castelnuovo bound. Standard textbooks give the Calstelnuovo bound for the genus g(C) of smooth nondegenerate curves C ⊂ ℙ^{3}, see [Ha; Theorem IV.6.4, page 351], for example. See also [ACGH], page 116). However the whole argument is valid for reduced, irreducible, local complete intersection curves such as the ones under consideration here. Indeed, the proof uses the Riemann-Roch theorem for the curve C, and it uses a General Position Lemma, to the effect that the section of C by a sufficiently general plane in ℙ^{3} consists of d distinct points, no three of which are on a line. The General Position Lemma lemma is valid for singular (reduced, irreducible) curves ([ACFH; page 109]; se also [Ha2] and references therein – this reference was pointed out to me by D. Perrin). A proof of the Riemann-Roch theorem for (possibly singular) curves given as divisors on a surface (such curves are automatically local complete intersections) may be found in [AK], Chapter VII, Section 1 (combine theorem (1.4), theorem (1.15) and Remark (1.17)).

Proposition 3 If for each nonsingular surface X ⊂ ℙ^{3} of degree n the number of curves C of degree d lying on X is finite, then there is an integer N(n, d) such that any nonsingular surface X ⊂ ℙ^{3} of degree n contains at most N(n, d) curves of degree d.

Let W ⊂ ℙ(H^{0}(ℙ^{3}, O(n))) be the open set corresponding to nonsingular surfaces of degree n. Let Z ⊂ G × W be the closed set whose points are pairs of points (c, f) with c corresponding to an integral curve C of degree d lying on the surface X defined by f. The projection map Z → W is a proper morphism. By hypothesis, any fibre of this morphism is finite, i.e. the morphism Z → W is quasi-finite. Being both proper and quasi-finite, the morphism Z → W is finite, see [Ha; exercise III.11.2, page 280]. This implies the existence of an integer N such that any fibre has at most N points.

Remark. Computing dimensions, one sees that for n big enough with respect to the degree, the general surface of degree n contains no (reduced, irreducible) curve of degree d at all.

Gathering the three propositions together, we conclude as follows.

Theorem For each pair n, d of positive integers with n > d+1, there exists an integer N(n, d) such that for any smooth projective surface X ⊂ ℙ^{3} of degree n, there are at most N(n, d) reduced and irreducible curves of degree d lying on X.

Remark. From this one may conclude an analogous result where one omits the condition ‘reduced and irreducible’. Indeed an effective Cartier divisor C ⊂ X of degree d in ℙ^{3} defines a divisor ∑_{i} n_{i}C_{i} with n_{i} > 0, where each C_{i} is an integral curve of degree d_{i}, and the sum ∑_{i} n_{i}d_{i} is equal to d.

References

[1] A. Altman, S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics 146, Springer-Verlag, 1970.

[2] E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of algebraic curves, Vol. I. (Springer, Berlin, 1985).

[3] R. Hartshorne, Algebraic geometry, (Springer, Berlin, 1977).

[4] R. Hartshorne, The genus of space curves, Ann. Univ. Ferrara - Sez. VII - Sc. Mat. vol. XL (1994) 207-223.