GEOMETRIC INVARIANT THEORY FOR HOLOMORPHIC FOLIATIONS ON ℂℙ² OF DEGREE 2

Abstract

Abstract. Let F 2 be the space of the holomorphic foliations on ‫ސރ‬ 2 of degree 2. In this paper we study the linear action PGL(3, ‫)ރ‬ × F 2 → F 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 2.2000 Mathematics Subject Classification. Primary 37F75, 14L24. Introduction.In this paper we study the properties of holomorphic foliations through the Geometric Invariant Theory (GIT), which was mainly developed by Hilbert and Mumford (see [6]).The Geometric Invariant Theory tells us that it is possible to study the action of a reductive group G on a projective variety V by stratifying the points of variety in two categories: unstable points and semistable points. By restricting the action of G to the semistable points we obtain what is called a good quotient. The set of semistable points contains the open set of stable points and the restriction of the action to the stable points gives us a geometric quotient.In most of the cases the variety V consists of certain geometric objects, such as algebraic curves or hypersurfaces. Furthermore, the usual action of G on V is such that objects are in the same orbit if and only if they are isomorphic.The unstable points form a Zariski closed set in V and are in some sense degenerate objects. For example, if we consider the natural action of PGL(3, ‫)ރ‬ on ‫ސރ‬ 9 , where ‫ސރ‬ 9 is the space of plane curves of degree 3, then a cubic plane curve is unstable if and only if it has a triple point, or a cusp, or two components tangent at a point. Another example is the action of PGL(2, ‫)ރ‬ in the space of binary forms of degree d. In this case a binary form of degree d is semistable if and only if it has no root of multiplicity greater that d 2 (see [13]).

Stratification of the space of foliations on CP2

“…In the first one we characterize the foliations with isolated singularities in the unique open stratum S 0 , which is the set of semistable foliations. In the second one we characterize the semistable foliations of degree 1, the proof of this can be found in Alcántara (2011).…”

Invariant theory of foliations of the projective plane

“…It was using this criterion that Goméz-Mont and Kempf [8] have shown that a foliation whose all singular points have Milnor number 1 is stable, that is, corresponds to a stable point of F m . And Alcántara [1], [2] has characterized the semi-stable foliations of degrees 1 and 2.…”

“…(In fact, they showed the same result holds for singular foliations of higher dimension spaces as well.) Only recently, Alcántara [1], [2] has characterized the semi-stable foliations of degree 1 and 2, and studied their quotient spaces.…”