Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-23T02:57:48.746Z Has data issue: false hasContentIssue false
  • English
  • Français

A cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces

Published online by Cambridge University Press:  03 July 2013

MIHAI HALIC  and
ROSHAN TAJAROD
MIHAI HALIC
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and MineralsDhahran 31261, Saudi Arabia. e-mail: mihai.halic@gmail.com
ROSHAN TAJAROD
Affiliation:
School of Mathematics, IPM, P.O. Box 19395-5746, Tehran, Iran. e-mail: roshan.tajarod@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we obtain a cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces with cyclic Picard group, which is similar to Horrocks' splitting criterion for locally free sheaves on projective spaces. We also recover a duality property which identifies a general K3 surface with a certain moduli space of stable sheaves on it, and obtain examples of stable, arithmetically Cohen–Macaulay, locally free sheaves of rank two on general surfaces of degree at least five in ${\mathbb P}^3$.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 517 - 527
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Auslander, M. and Buchsbaum, D.Homological dimension in local rings. Trans. Amer. Math. Soc. 85 (1957), 390–405.CrossRefGoogle Scholar
[2]Ballico, E.Rank 2 totally arithmetically Cohen–Macaulay vector bundles on an abelian surface with $\rm{Num}(X)\cong{\mathbb Z}$. Rend. Istit. Mat. Univ. Trieste 40 (2008), 45–53.Google Scholar
[3]Ballico, E.Rank 2 arithmetically Cohen–Macaulay vector bundles on K3 and Enriques surfaces. Albanian J. Math. 3 (2009), 3–11.CrossRefGoogle Scholar
[4]Ballico, E.Weakly ACM vector bundles on ruled surfaces and their blowing-ups. Int. J. Pure Appl. Math. 51 (2009), 11–17.Google Scholar
[5]Brodmann, M. and Sharp, R.Local Cohomology: An algebraic introduction with geometric applications (Cambridge University Press, 1998).CrossRefGoogle Scholar
[6]Chiantini, L. and Faenzi, D.Rank 2 arithmetically Cohen–Macaulay bundles on a general quintic surface. Math. Nachr. 282 (2009), 1691–1708.CrossRefGoogle Scholar
[7]Deligne, P. and Katz, N.Groupes de monodromie en géométrie algébrique (SGA 7 II). Lecture Notes in Math. vol. 340 (Springer-Verlag, 1973).CrossRefGoogle Scholar
[8]Hartshorne, R.Algebraic Geometry. 8th Ed. (Springer-Verlag, New York, 1997).Google Scholar
[9]Huybrechts, D. and Lehn, M.The geometry of moduli spaces of sheaves. 2nd Ed. (Cambridge University Press, 2010).CrossRefGoogle Scholar
[10]Horrocks, G.Vector bundles on the punctured spectrum of a local ring. Proc. London University Math. Soc. 14 (1964), 689–713.CrossRefGoogle Scholar
[11]Langer, A.Semistable sheaves in positive characteristic. Ann. Math. 159 (2004), 251–276.CrossRefGoogle Scholar
[12]Matsumura, H.Commutative algebra. 2nd Ed. (The Benjamin/Cummings Publishing Company, 1980).Google Scholar
[13]Mukai, S.Duality of polarized K3 surfaces. in New Trends in Algebraic Geometry, Hulek, K.et al. (eds.) Lond. Math. Soc. vol. 264 (Cambridge University Press, 1999), pp. 311–326.CrossRefGoogle Scholar
[14]Oda, T.Vector bundles on abelian surfaces. Invent. Math. 13 (1971), 247–260.CrossRefGoogle Scholar
[15]Ravindra, G.V. and Srinivas, V.The Grothendieck–Lefschetz theorem for normal projective varieties. J. Algebraic Geom. 15 (2006), 563–590.CrossRefGoogle Scholar
[16]Ravindra, G.V. and Srinivas, V.The Noether–sLefschetz theorem for the divisor class group. J. Algebra 322 (2009), 3373–3391.CrossRefGoogle Scholar
[17]Saint-Donat, B.Projective models of K3 surfaces. Amer. J. Math. 96 (1974), 602–639.CrossRefGoogle Scholar
[18]Schwarzenberger, R.Vector bundles on algebraic surfaces. Proc. London Math. Soc. 11 (1961), 601–622.CrossRefGoogle Scholar