The Geometric Invariant Theory tells that it is possible to study the action of a reductive
group G on a projective variety V by stratifying the points of V in two categories: unstable
points and semistable ones. The unstable points form a Zariski closed set in V and are in
some sense degenerate objects. The restriction of the action of G to the semistable points
yields to a good quotient. The set of semistable points contains the open set of stable
points and the restriction of G to this set gives a geometric quotient.
Let us consider the situation where V is the space of holomorphic foliations of degree
d on Pn (C) and the group PGL(n + 1, C) of automorphisms of Pn (C) acts by change of
coordinates. Gómez-Mont and Kempf proved that a foliation with only non-degenerate
singularities is stable ([7]).
In the article under review the author considers the space F2 of holomorphic foliations
on the complex projective plane of degree 2. More precisely she studies the linear action
Aut(P2 (C))×F2 → F2 , (g, F ) 7→ DgF ◦g −1 in the sense of the Geometric Invariant Theory.
In Section 4 she obtains the generators of unstable foliations, then gives a characterisation
of unstable, semistable but not stable and stable folitations according to the multiplicity,
the Milnor number of its singular points and the existence of invariant lines.
Theorem. Let F be an element of F2 with isolated singularities. Then F is unstable if
and only if exactly one of the following holds:
• there exists a singular point of multiplicity 2;
• the foliation F has an invariant line with a unique singular point with multiplicity
1 and Milnor number 5.
Theorem. Let F be an element of F2 with isolated singularities. Then f is semistable,
but not stable, if and only if the multiplicity of every singular point is 1, and F has an
invariant line L with one singular point with Milnor number 3 or 4 or L has two different
singular points, one of them with Milnor number 3, 4, 5 or 6.
Corollary. A foliation F ∈ F2 with isolated singularities is stable if and only if every
singular point of F has multiplicity 1 and if one the following holds:
• F does not have invariant lines;
• if L is an invariant line for F then
– either L has two different singularities, one with Milnor number 2 and the
other with Milnor number 1;
– or L has two different singularities, both with Milnor number 2;
– or L has three different singularities.
In Section 5 the author shows that a generic unstable foliation F of degree 2 has the following property: there exists a rational fibration on P1 (C) whose generic fibre is transverse
to the leaves of the foliation. In other words F is a Ricatti foliation.
Finally in the last section the author proves that the geometric quotient of non-degenerate
holomorphic foliations on P2 (C) of degree 2 without invariant lines is the coarse moduli
space of the polarised del Pezzo surfaces of degree 2.
1
2
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