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.