Weil's proof of the Riemann Hypothesis for projective curves relies upon the following positivity result: Let $\mathbb{F}q$ be the finite field with $q$ elements, $\overline{\mathbb{F}q}$ its closure, and $X$ a projective curve 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 the 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 for projective varieties in general the sole reason that Weil proof does not generalise to all projective varieties?