Q. Would it be possible for you to tell me what the
precise statement of Castelnuovo's base point free
pencil trick is? If there isn't actually a statement,
would you mind telling me what exactly it allows us to
deduce? I think we used it to deduce Max Noether's
theorem and Clifford's theorem, but I couldn't pinpoint
it and it seems important.
====
The Castelnuovo free pencil trick.
|D| is a _free pencil_ (or a g^1_d) if s1, s2 in L(C,D)
have div s1 + D = D1 and div s2 + D = D2, where D1, D2
have no common support. (They are effective divisors by
definition of the RR space L(C,D)).
Theorem.
Assuming the above, for any divisor A, the two products
s1*L(C,A) and s2*L(C,A)
(all multiplications take place in k(C)) are subspaces of
L(C,A+D),
and their intersection is exactly the set of (s1*f = s2*g)
where f = s2*c, g = s1*c for c in L(C,A-D).
Proof.
In fact, c = g/s1 = f/s2 in k(C) is well defined. Consider
its divisor div c. On the one hand since f, g in L(C,A),
both div f + A >= 0, div g + A >= 0.
Therefore
div c + A = (div f + A) - div s2 >= D - D2
and in the same way,
div c + A = (div g + A) - div s1 >= D - D1
However, D1 and D2 are effective divisors supported at
disjoint sets of points, so at every point of C the
multiplicity of (div c + A) is >= the multiplicity of D.
Therefore c in L(C, A-D). QED
This was used in Part 4, page 6 in the proof that s1, s2
in the complete sections ring R(C,D) form a regular sequence.
[I also mentioned it in lectures in the proof of Max Noether's
theorem that for nonhyperelliptic curves,
H^0(C, KC) x H^0(C, KC) ->> H0(C, 2KC)
is surjective (but that didn't make it into the lecture
notes, see next Addendum).]
It was not really used in that form in Clifford's theorem,
(Part 3, Ex. 3), which also depended on multiplication
L(C,D) x L(C,K-D) -> L(C,K), but is an easier formal
argument.