On a field of positive characteristic we have the following well-known and important result:
(Hodge Index theorem) Let $H$ be an ample divisor on a surface $X$, and suppose that $D$ is a divisor, $D \not \equiv 0$, with $D.H=0$. Then $D^2<0$.
It was independently proven in:
Beniamino Segre, Intorno ad un teorema di Hodge sulla teoria della base per la curve di una superficie algebraic (1937)
Alexander Grothendieck, Sur une note de Mattuck-Tate (1958)
My question is:
Are those proofs essentially the same as the one on Hartshorne's Algebraic Geomtry book? (chapter V, section 1)