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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)

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