Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T17:25:19.624Z Has data issue: false hasContentIssue false

The resolution property of algebraic surfaces

Published online by Cambridge University Press:  09 November 2011

Philipp Gross*
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Deutschland, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Art71]Artin, M., On the joins of Hensel rings, Adv. Math. 7 (1971), 282296.Google Scholar
[BGI71]Berthelot, P., Grothendieck, A. and Illusie, L. (eds), Théorie des intersections et théorème de Riemann-Roch: Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, 1971), Dirigé par Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre.Google Scholar
[Bor63]Borelli, M., Divisorial varieties, Pacific J. Math. 13 (1963), 375388.Google Scholar
[Bou65]Bourbaki, N., Éléments de mathématique, in Fasc. XXXI. Algèbre commutative, Actualités Scientifiques et Industrielles, vol. 1314 (Hermann, Paris, 1965), Chapitre 7: Diviseurs.Google Scholar
[BS03]Brenner, H. and Schröer, S., Ample families, multihomogeneous spectra, and algebraization of formal schemes, Pacific J. Math. 208 (2003), 209230.CrossRefGoogle Scholar
[BV75]Bruns, W. and Vetter, U., Die Verallgemeinerung eines Satzes von Bourbaki und einige Anwendungen, Manuscripta Math. 17 (1975), 317325.Google Scholar
[Con07]Conrad, B., Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), 205257.Google Scholar
[EG85]Evans, E. G. and Griffith, P., Syzygies, London Mathematical Society Lecture Note Series, vol. 106 (Cambridge University Press, Cambridge, 1985).Google Scholar
[Fer03]Ferrand, D., Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), 553585.Google Scholar
[Gro61a]Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci. 8 (1961), 1222.Google Scholar
[Gro61b]Grothendieck, A., Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Publ. Math. Inst. Hautes Études Sci. 11 (1961), 349511.Google Scholar
[Gro65]Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 1231.Google Scholar
[Gro66]Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 1255.Google Scholar
[GD71]Grothendieck, A. and Dieudonné, J. A., Éléments de géométrie algébrique. I, Die Grundlehren der mathematischen Wissenschaften, vol. 166 (Springer, Berlin–Heidelberg–New York, 1971).Google Scholar
[Ill05]Illusie, L., Grothendieck’s existence theorem in formal geometry, in Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123 (American Mathematical Society, Providence, RI, 2005), 179, 233 with a letter (in French) by Jean-Pierre Serre.Google Scholar
[Jel05]Jelonek, Z., On the projectivity of threefolds, Proc. Amer. Math. Soc. 133 (2005), 25392542 (electronic).CrossRefGoogle Scholar
[Kle66]Kleiman, S. L., Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293344.CrossRefGoogle Scholar
[Mum70]Mumford, D., Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5 (Published for the Tata Institute of Fundamental Research, Bombay, 1970).Google Scholar
[Pay09]Payne, S., Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom. 18 (2009), 136.CrossRefGoogle Scholar
[Ray70]Raynaud, M., Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, vol. 119 (Springer, Berlin, 1970).Google Scholar
[RV04]Roth, M. and Vakil, R., The affine stratification number and the moduli space of curves, in Algebraic structures and moduli spaces, CRM Proceedings and Lecture Notes, vol. 38 (American Mathematical Society, Providence, RI, 2004), 213227.CrossRefGoogle Scholar
[Sch99]Schröer, S., On non-projective normal surfaces, Manuscripta Math. 100 (1999), 317321.Google Scholar
[SV04]Schröer, S. and Vezzosi, G., Existence of vector bundles and global resolutions for singular surfaces, Compos. Math. 140 (2004), 717728.CrossRefGoogle Scholar
[Sch82]Schuster, H.-W., Locally free resolutions of coherent sheaves on surfaces, J. Reine Angew. Math. 337 (1982), 159165.Google Scholar
[Som78]Sommese, A. J., Submanifolds of Abelian varieties, Math. Ann. 233 (1978), 229256.CrossRefGoogle Scholar
[Ste98]Steffen, F., A generalized principal ideal theorem with an application to Brill–Noether theory, Invent. Math. 132 (1998), 7389.Google Scholar
[Tot04]Totaro, B., The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 122.Google Scholar
[Tot10]Totaro, B., Line bundles with partially vanishing cohomology (2010), arXiv:1007.3955v1 [math.AG].Google Scholar
[Voi02]Voisin, C., A counterexample to the Hodge conjecture extended to Kähler varieties, Int. Math. Res. Not. IMRN 20 (2002), 10571075.Google Scholar
[W{{{\relax ł}}}o93]Włodarczyk, J., Embeddings in toric varieties and prevarieties, J. Algebraic Geom. 2 (1993), 705726.Google Scholar
[W{{{\relax ł}}}o99]Włodarczyk, J., Maximal quasiprojective subsets and the Kleiman–Chevalley quasiprojectivity criterion, J. Math. Sci. Univ. Tokyo 6 (1999), 4147.Google Scholar