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On the singular scheme of codimension one holomorphic foliations in P(3)

Giraldo Suárez, Luis and Pan-Collantes , Antonio J. (2010) On the singular scheme of codimension one holomorphic foliations in P(3). International Journal of Mathematics, 21 (7). pp. 843-858. ISSN 0129-167X

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Abstract

In this work, we begin by showing that a holomorphic foliation with singularities is reduced if and only if its normal sheaf is torsion-free. In addition, when the codimension of the singular locus is at least two, it is shown that being reduced is equivalent to the reflexivity of the tangent sheaf. Our main results state on one hand, that the tangent sheaf of a codimension one foliation in P3 is locally free if and only if the singular scheme is a curve, and that it splits if and only if it is arithmetically Cohen–Macaulay. On the other hand, we discuss when a split foliation in P3 is determined by its singular scheme.


Item Type:Article
Uncontrolled Keywords:Holomorphic foliations; reflexive sheaves; split vector bundles
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:16332
References:

P. Baum and R. Bott, Singularities of holomorphic foliations, J. Differential Geom.7 (1972) 279–342.

O. Calvo-Andrade, Irreducible components of the space of holomorphic foliations,Math. Ann. 29(1) (1994) 751–767.

O. Calvo-Andrade, D. Cerveau, L. Giraldo and A. Lins-Neto, Irreducible components of the space of foliations associated to the affine Lie algebra, Ergodic Theory Dynam. Systems 24(4) (2004) 987–1014.

A. Campillo and J. Olivares, Polarity with respect to a foliation and Cayley-Bacharach Theorems, J. Reine Angew. Math. 534 (2001) 95–118.

D. Cerveau and A. Lins-Neto, Irreducible components of the space of holomorphic foliations of degree two in CP(n), n ≥ 3, Ann. Math. 143 (1996) 577–612.

F. Cukierman and J. V. Pereira, Stability of holomorphic foliations with split tangent sheaf, Amer. J. Math. 130(2) (2008) 413–439.

F. Cukierman, M. G. Soares and I. Vainsencher, Singularities of logarithmic foliations, Compos. Math. 142 (2006) 131–142.

D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, Vol. 150, Springer Verlag (1995).

X. Gómez-Mont, Universal families of foliations by curves, Ast´erisque 150–151 (1987) 109–129.

G.-M. Greuel, G. Pfister and H. Schönemann, Singular 3.0, a computer algebra system for polynomial computations, Center for Computer Algebra, University of Kaiserslautern (2005), http://www.singular.uni-kl.de.

R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, Vol. 52 (Springer, 1977).

R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980) 121–176.

F. Hirzebruch, Topological Methods in Algebraic Geometry, Classics in Mathematics (Springer-Verlag, Berlin, 1995).

Deposited On:12 Sep 2012 10:59
Last Modified:07 Feb 2014 09:27

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