The lemma of Castelnuovo-Mumford.

Let's begin with a definition:

Definition: A coherent sheaf, F on Pn is said to be m-regular if hi(F(m-i))=0, for any i>0.

Lemma:(Castelnuovo-Mumford) If F is m-regular then:

  1. hi(F(k))=0 if k > m-i, i>0
  2. F(k) is generated by global sections for k > m-1
  3. The graded module H0*(F) is generated by its elements of degree <(m+1)

The proof (which is not difficult) is by induction on n (cf Mumford: "On curves on an algebraic surface" p.99)

We shall use this lemma essentially for F = IC, the ideal sheaf of a curve C in P3. The vanishing of h2(IC(m-2))= h1(OC(m-2)) will follow (in general) just from considerations on the degree. The vanishing h1(IC(m-1))=0 is (in general) much more difficult to obtain.

If IC is m-regular we have:
  1. hi(IC(k))=0 if k > m-i, i>0
  2. IC(k) is generated by global sections if k > (m-1).
  3. The homogeneous ideal I(C) is generated by its elements of degree < (m+1).

An interesting consequence of the fact that IC(k) is generated by global sections is:

Lemma: Let C in P3 be a smooth irreducible curve, if IC(k) is generated by global sections, then the general surface of degree k containing C is smooth.

 

Proof: Since the intersection of the surfaces in |H0(IC (k))| is C, this linear system has no base-points outside of C. By Bertini's theorem the general element of this linear system is smooth outside of C. Now consider the exact sequence:

0 -> IC2(k) -> IC(k) -> NC*(k) -> 0

This exact sequence shows that the sections generating IC(k) yield sections generating NC*(k); but if a vector space, V, of global sections of a rank two bundle E on a curve C generate E, then the general section in V does not vanish on C (as a vector bundle section) (because it vanishes in the expected codimension which here is 2). In our case this precisely means that there is a degree k surface containing C which is smooth along C.

Remark: Since IC(k) is generated by global sections if k >> 0 (Serre's theorem), this shows that every smooth curve C in P3 is contained in a smooth surface.