Improvement of Grauert-Riemenschneider's theorem for a normal surface
Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 13-23.

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. V0 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.

Soit X ˜ une désingularisation d’une surface normale X. Le groupe Pic(X ˜) est muni d’une relation d’ordre L ̲0, définie par L. V0 pour tout diviseur effectif exceptionnel V. En comparant avec la relation d’ordre usuelle, nous définissons le plafond de L qui est un diviseur exceptionnel. Cette notion permet d’améliorer le théorème usuel d’annulation et d’en déduire un critère de rationalité et une formule pour le genre d’une courbe sur une surface normale; la difficulté réside dans le cas d’un diviseur de Weil qui n’est pas un diviseur de Cartier.

@article{AIF_1982__32_4_13_0,
     author = {Giraud, Jean},
     title = {Improvement of {Grauert-Riemenschneider's} theorem for a normal surface},
     journal = {Annales de l'Institut Fourier},
     pages = {13--23},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {32},
     number = {4},
     year = {1982},
     doi = {10.5802/aif.892},
     mrnumber = {84f:14025},
     zbl = {0488.32013},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.892/}
}
TY  - JOUR
TI  - Improvement of Grauert-Riemenschneider's theorem for a normal surface
JO  - Annales de l'Institut Fourier
PY  - 1982
DA  - 1982///
SP  - 13
EP  - 23
VL  - 32
IS  - 4
PB  - Imprimerie Durand
PP  - 28 - Luisant
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.892/
UR  - https://www.ams.org/mathscinet-getitem?mr=84f:14025
UR  - https://zbmath.org/?q=an%3A0488.32013
UR  - https://doi.org/10.5802/aif.892
DO  - 10.5802/aif.892
LA  - en
ID  - AIF_1982__32_4_13_0
ER  - 
%0 Journal Article
%T Improvement of Grauert-Riemenschneider's theorem for a normal surface
%J Annales de l'Institut Fourier
%D 1982
%P 13-23
%V 32
%N 4
%I Imprimerie Durand
%C 28 - Luisant
%U https://doi.org/10.5802/aif.892
%R 10.5802/aif.892
%G en
%F AIF_1982__32_4_13_0
Giraud, Jean. Improvement of Grauert-Riemenschneider's theorem for a normal surface. Annales de l'Institut Fourier, Volume 32 (1982) no. 4, pp. 13-23. doi : 10.5802/aif.892. https://aif.centre-mersenne.org/articles/10.5802/aif.892/

[1] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math., (1966), 129-136. | MR: 33 #7340 | Zbl: 0142.18602

[2] L. Badescu, Dualizing divisors of two dimensional singularities, Rev. Roum. Math. Pures et Appl., XXV, 5, 695-707. | MR: 82c:14029 | Zbl: 0457.14017

[3] J. Giraud, Intersections sur les surfaces normales, Séminaire sur les singularités des Surfaces, Janv. 1979, École Polytechnique. | Zbl: 0699.14011

[4] H. Grauert, O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegrupper auf komplexen Raümen, Inv. Math., 11 (1970), 263-292. | MR: 46 #2081 | Zbl: 0202.07602

[5] J. Lipman, Rational singularities, Pub. Math. I.H.E.S., 36 (1969), 195-279. | Numdam | Zbl: 0181.48903

[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. | Numdam | MR: 27 #3643 | Zbl: 0108.16801

[7] J. Wahl, Vanishing theorems for resolutions of surface singularities, Inv. Math., 31 (1975), 17-41. | MR: 53 #13225 | Zbl: 0314.14010

Cited by Sources: