Stratification of the space of foliations on CP2

Abstract

We describe an algorithm for constructing an algebraic stratification of the space of holomorphic foliations on CP 2 of degree d with respect to the action of Aut(CP 2 ) by change of coordinates. The strata are non-singular, locally-closed algebraic varieties. We show that these varieties parameterize foliations with certain type of degenerate singular points. We give the explicit form of the foliation in some strata. We also obtain the dimension of these varieties.

Classification of Foliations on CP^2 of degree 3 with Degenerate Singularities

“…We also get the dimension and explicit generators for each stratum. Similar results for degree 2 are given in [2] and in [3], we have some general results for degree d.…”

“…Norbert A'Campo and Vladimir Popov give in [15] a computer program such that given a reductive group and one of its representation, the output is the finite subset B of virtual 1-parameter subgroups for the above stratification. For a more detailed construction of the virtual 1-parameter subgroups in the case of the action by change of coordinates of SL(3, C) in F d we refer to section 3 of [3]. For F 3 the virtual 1-parameter subgroups for the stratification are: Now we consider the induced representation H 0 (CP 2 , T CP 2 (2)) of the Lie algebra sl 3 (C).…”

“…If X ∈ Y ss 16 then b 1,0 = 0 and c 0,3 = 0. We have In [3] we also have studied the strata S 6 , S 9 and S 16 . As a consequence of the above we can mention the following general result.…”

Stability of pencils of plane sextics and Halphen pencils of index two

“…A priori, for each choice of coordinates in P 2 one would need to test all possible values of a ∈ [−1/2, 1] ∩ Q to verify the stability criterion. Because the function (for a fixed P and a choice of coordinates) (1) µ(P, λ) . = min{e ijkl (a) :…”

“…In [10], the authors give a criterion for when a foliation of P 2 of degree seven is a Halphen pencil of index two. In [1], a stratification of the space of foliations on P 2 of degree d is given (in particular d = 7), thus providing an alternative description of the semistable and unstable Halphen pencils of index two under projective equivalence -viewed as foliations. However, such stratification does not provide a description of the stable pencils, which we do in this paper.…”