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

Jean Giraud

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 4, page 13-23
  • ISSN: 0373-0956

Abstract

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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 divisor.

How to cite

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Giraud, Jean. "Improvement of Grauert-Riemenschneider's theorem for a normal surface." Annales de l'institut Fourier 32.4 (1982): 13-23. <http://eudml.org/doc/74556>.

@article{Giraud1982,
abstract = {Let $\widetilde\{X\}$ be a desingularization of a normal surface $X$. The group Pic$(\widetilde\{X\})$ is provided with an order relation $L \underline\{\gg \} 0$, defined by $L$. $V\le 0$ 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 divisor.},
author = {Giraud, Jean},
journal = {Annales de l'institut Fourier},
keywords = {effective exceptional divisor; vanishing theorem; numerical criterion for rationality; genus formula; Weil divisor; Cartier divisor; Picard group},
language = {eng},
number = {4},
pages = {13-23},
publisher = {Association des Annales de l'Institut Fourier},
title = {Improvement of Grauert-Riemenschneider's theorem for a normal surface},
url = {http://eudml.org/doc/74556},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Giraud, Jean
TI - Improvement of Grauert-Riemenschneider's theorem for a normal surface
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 4
SP - 13
EP - 23
AB - Let $\widetilde{X}$ be a desingularization of a normal surface $X$. The group Pic$(\widetilde{X})$ is provided with an order relation $L \underline{\gg } 0$, defined by $L$. $V\le 0$ 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 divisor.
LA - eng
KW - effective exceptional divisor; vanishing theorem; numerical criterion for rationality; genus formula; Weil divisor; Cartier divisor; Picard group
UR - http://eudml.org/doc/74556
ER -

References

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  1. [1] M. ARTIN, On isolated rational singularities of surfaces, Amer. J. Math., (1966), 129-136. Zbl0142.18602MR33 #7340
  2. [2] L. BADESCU, Dualizing divisors of two dimensional singularities, Rev. Roum. Math. Pures et Appl., XXV, 5, 695-707. Zbl0457.14017MR82c:14029
  3. [3] J. GIRAUD, Intersections sur les surfaces normales, Séminaire sur les singularités des Surfaces, Janv. 1979, École Polytechnique. Zbl0699.14011
  4. [4] H. GRAUERT, O. RIEMENSCHNEIDER, Verschwindungssätze für analytische Kohomologiegrupper auf komplexen Raümen, Inv. Math., 11 (1970), 263-292. Zbl0202.07602MR46 #2081
  5. [5] J. LIPMAN, Rational singularities, Pub. Math. I.H.E.S., 36 (1969), 195-279. Zbl0181.48903
  6. [6] D. MUMFORD, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Pub. Math. I.H.E.S., 11 (1961), 229-246. Zbl0108.16801MR27 #3643
  7. [7] J. WAHL, Vanishing theorems for resolutions of surface singularities, Inv. Math., 31 (1975), 17-41. Zbl0314.14010MR53 #13225

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