Complete intersections in projective spaces

Bas Edixhoven

June 30, 1999

The aim of this appendix is to provide a proof of the following theorem. The proof is very standard, and the theorem is probably known (at least implicitly) by those who compute the cohomology of twisted ideal sheaves of complete intersections in projective spaces.

Theorem 1 Let S be a complex analytic variety, with a given point s. Let n ≥ 1 be an integer, and X → S be a smooth analytic family of projective complex analytic varieties of dimension d > 0, embedded in ℙ_{S}^{n} := ℙ^{n}(ℂ) × S. Suppose that the fibre X_{s} at s is a complete intersection. Then there is an open neighborhood of s in S over which X is a complete intersection.

Proof. Let I be the ideal sheaf of X in the structure sheaf O of ℙ_{S}^{n}. Let f: ℙ_{S}^{n} → S denote the projection, let c := n − d, and r_{1}, …, r_{c} the degrees of a set of hypersurfaces H_{i} in ℙ_{s}^{n} of which X_{s} is the intersection. Each H_{i} is the zero locus of a section of O_{ℙ_{s}^{n}}(r_{i}). All we have to show is that the sections defining the H_{i} can be lifted to sections of f_{∗}I(r_{i}) over some neighborhood of s, since then X is contained in the complete intersection defined by these lifts, hence equal to it. We will show in fact that f_{∗}I(r) is free as O_{S}-module on a neighborhood of s for each integer r.

In order to show that f_{∗}I(r) is locally free at s, it is enough, by the standard results on base change and coherent cohomology (see [2, III, Thm. 12.11] for the case of schemes), to show that H^{1}(ℙ_{s}^{n}, I_{s}(r)) is zero, with I_{s} = I/m_{s}I and m_{s} the ideal of s in O_{S}. Since the holomorphic tangent map of X → S is surjective, O_{X} is flat over O_{S}, hence tensoring the short exact sequence:

0 ⟶ I(r) ⟶ O(r) ⟶ O_{X}(r) ⟶ 0

over O_{S} with ℂ = O_{S}/m_{s} gives a short exact sequence:

0 ⟶ I_{s}(r) ⟶ O_{s}(r) ⟶ O_{X_{s}}(r) ⟶ 0.

This shows that I_{s}(r) is the rth twist of the ideal sheaf of X_{s} in ℙ_{s}^{n}. The hypothesis that X_{s} is a complete intersection, with degrees r_{1}, …, r_{c}, gives a surjection of O_{s}-modules:

⨁_{i=1}^{c} O_{s}(r − r_{i}) ⟶ I_{s}(r).

This surjection can then be extended to a Koszul type resolution (see for example [1, 5.3]):

K ⟶ I_{s}(r) ⟶ 0,

with K a complex (placed in degrees ≤ 0) such that, for p ≤ 0:

K^{p} = (⋀^{1−p} (⨁_{i=1}^{c} O_{s}(−r_{i}))) (r).

In particular, K^{p} = 0 for p > 0 and for p ≤ −c. We view this resolution as a quasi-isomorphism from the complex K to the complex consisting of just I_{s}(r), placed in degree 0. Then it follows that the cohomology of I_{s}(r) is the hypercohomology of K, and that there is a spectral sequence E with E_{1}^{p,q} = H^{q}(ℙ_{s}^{n}, K^{p}), converging to the cohomology of I_{s}(r) (see for example [1, 3.5]). Since each K^{p} is a direct sum of invertible O_{s}-modules, we have E_{1}^{p,q} = 0 for q not equal to 0 or n (see for example [2, III, Thm. 5.1]; this is an interesting property of projective spaces). Hence the E_{1}-term has the following form:

H^{n}(K^{1−c}) → ⋯ → H^{n}(K^{0})
0  → ⋯ →  0
⋮  ⋮
0  → ⋯ →  0
H^{0}(K^{1−c}) → ⋯ → H^{0}(K^{0})

It follows that E_{1}^{p,q} = 0 for all (p, q) with p + q = 1, hence that E_{r}^{p,q} = 0 for all r ≥ 1 and such (p, q). (Note that we have used here that d > 0, which is easily seen to be a necessary condition.) To finish the proof, note that H^{1}(ℙ_{s}^{n}, I_{s}(r) is the direct sum of the E_{∞}^{p,q} with p + q = 1, hence is zero. □

Remark 2 As the proof shows, the hypothesis that X → S be smooth can be replaced by the one that X → S be flat. The proof works also in the contexts of analytic spaces (not necessarily reduced) and schemes. The spectral sequence above degerates at E_{2}, and can be used to compute the cohomology groups of I_{s}. This last fact makes it very plausible that this computation is not new.

References

[1] P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley Interscience, 1978.

[2] R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics 52. Springer-Verlag, 1977.