This exercise is intended to show that Num$X$ (divisors modulo numerical equivalence) on a smooth, projective, integral surface $X$ is a finitely generated free abelian group, without using some big theorem of Neron and Severi. We assume the ground field is algebraically closed of characteristic 0.

The first part of the question asks to show that the map $c:Pic(X) \rightarrow H^1(X,\Omega_X)$ induced by $\mathcal{O}_X^* \rightarrow \Omega_X$, $f\rightarrow f^{-1}df$ is compatible with the pairings on $Pic(X)$ (intersection product) and $H^1(X,\Omega_X)$ (Serre duality). I'm not convinced that this is as trivial as certain online solutions make it out to be, but fine, I'm willing to accept that this is true by naturalness of Serre duality.

What I'm interested in is the second part of the question, which says to conclude that Num$X$ is finitely generated and free. I can use the above to show that Num$X$ is an abelian subgroup of the finite dimensional vector space $H^1(X,\Omega_X)$, although this does not suffice. The two online solution references I've seen both cite the exponential sequence in Appendix B of Hartshorne, but this seems bogus to me and not at all what the author intended. For one, it's not clear to me how that will give the answer for say $k=\bar{\mathbb{Q}}$.

  • $\begingroup$ Have you solved the problem?. I am curious since (as you say) the online references cite the exponential sequence that seems to be different from the Hartshorne's hint. $\endgroup$ – Holonomia Feb 11 '16 at 13:41

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