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GEOMETRIC INVARIANT THEORY FOR HOLOMORPHIC FOLIATIONS ON ℂℙ2 OF DEGREE 2

Published online by Cambridge University Press:  22 December 2010

CLAUDIA R. ALCÁNTARA
CLAUDIA R. ALCÁNTARA*
Affiliation:
Departamento de Matemáticas, Universidad de Guanajuato, Callejón Jalisco s/n, A.P. 402, C.P. 36000, Guanajuato, Gto. México; Université de Grenoble I, Département de Mathématiques, Institut Fourier. 38402 Saint-Martin d'Hères Cedex, France. e-mail: [email protected]
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Abstract

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Let 2 be the space of the holomorphic foliations on ℂℙ2 of degree 2. In this paper we study the linear action PGL(3, ℂ) × 2 → 2 given by gX = DgX ^(g−1) in the sense of the Geometric Invariant Theory. We obtain a characterisation of unstable and stable foliations according to properties of singular points and existence of invariant lines. We also prove that if X is an unstable foliation of degree 2, then X is transversal with respect to a rational fibration. Finally we prove that the geometric quotient of non-degenerate foliations without invariant lines is the moduli space of polarised del Pezzo surfaces of degree 2.

Keywords

Primary 37F7514L24
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 1 , January 2011 , pp. 153 - 168
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

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