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Improvement of Grauert-Riemenschneider's theorem for a normal surface

Jean Giraud — 1982

Annales de l'institut Fourier

Let X ˜ be a desingularization of a normal surface X . The group Pic ( X ˜ ) is provided with an order relation L ≫ _ 0 , defined by L . V ≤ 0 for any effective exceptional divisor V . Comparing to the usual order relation we define the ceiling of L which is an exceptional divisor. This notion allows us to improve the usual vanishing theorem and we deduce from it a numerical criterion for rationality and a genus formula for a curve on a normal surface; the difficulty lies in the case of a Weil divisor which is not a Cartier...

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