Does Weil's proof of the Weil conjecturesRiemann Hypothesis for projective curves work for FP^2, F arelies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, even though it is not$\overline{\mathbb{F}q}$ its closure, and $X$ a projective curve? What I think my question reduces to defined over $\overline{\mathbb{F}q}$. Let $D$ be a divisor class on $X \times X$ (where two divisors are related if they differ by a principal divisor), and let $\{p\} \times C$ and $C \times \{p\}$ denote the classes containing these elements. Define an integer valued mapping Tr on the divisor classes by
Tr$(D) = D \bullet (\{p\} \times C) + D \bullet (C \times \{p\}) - D \bullet \Delta$,
where $\bullet$ is whetherthe standard intersection product, and $\Delta$ is the diagonal divisor class. Now let $\circ$ be the multiplication induced on divisor classes by composition and let $D^t$ be the divisor given by the composition of $D$ and the permutation of the two factors of an element of $C \times C$. It can be shown that
Tr$(D \circ D^t) \geq 0$,
for all $D$. This result is usually called Castelnuovo Positivity, or Weil Positivity. My questions are as follows:
(1) Does anyone know of a good expository proof of this result available online?
(2) Does the result hold for any projective surfaces which are not curves? More specifically, does it hold for $\mathbb{C}P^2$ or for flag varieties?
(3) Is the failure of this Castelnuovo positivity holdsPositivity for projective varieties in this case.general the sole reason that Weil proof does not generalise to all projective varieties?
