UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIAS EXATAS DEPARTAMENTO DE MATEM ´ATICA
Viviana Ferrer Cuadrado
Enumerative Aspects of Holomorphic Foliations
Belo Horizonte
VIVIANA FERRER CUADRADO
Enumerative Aspects of Holomorphic Foliations
Tese apresentada ao Instituto de Matem´atica da Universidade Federal de Minas Gerais para obten¸c˜ao do t´ıtulo de Doutor em Matem´atica.
´
Area de Concentra¸c˜ao: Geometr´ıa Alg´ebrica.
Orientador: Israel Vainsencher.
Belo Horizonte
Ferrer Cuadrado, Viviana
F385e Enumerative Aspects of Holomorphic Foliations/Viviana Ferrer Cuadrado–Belo Horizonte, 2010.
110 f. :il. ; 29cm.
Tese(doutorado)–Universidade Federal de Minas Gerais. Orientador: Israel Vainsencher.
1.Geometr´ıa enumerativa - Teses. 2. Folhea¸c˜oes (Matem´atica) Teses. 3. -Folhea¸c˜oes Holomorfas- Teses. I. Orientador. II. T´ıtulo.
Agradecimentos
Agrade¸co a...
Israel, por toda a matem´atica que aprendi com ele, por seu jeito de fazer matem´atica, pelo apoio e confian¸ca que me deu, mas principalmente pela amizade.
Ao Cnpq pelo apoio financiero.
Ao Departamento de Matem´atica da UFMG, em particular ao grupo de Geometr´ıa Alg´ebrica e Folhea¸c˜oes, especialmente a Adriana e a Eden pela camaradagem.
Aos integrantes da banca, que com seus coment´arios e sugesto˜es tem ajudado a mel-horar este texto.
Olhando mais atr´as, agrade¸co `a UDELAR pela forma¸c˜ao que me deu e que facilitou muito meu caminho em esta etapa.
Olhando mais atras ainda, aos meus pais, pela vida que me deram.
Aos amigos que me acompanharam nesta etapa, e que fizeram minha vida mais f´acil em Belo Horizonte, Heleno, Luis e em especial a Reginaldo, Pablito e Luciano.
A todos meus amigos de cora¸c˜ao que estiveram longe fisicamente mas sempre perto. Especialmente `a Negra, por compartilhar o caminho.
“ Para luchar contra el pragmatismo y la horrible tendencia a la consecuci´on de fines ´utiles...”
Aspectos Enumerativos das Folhea¸c˜
oes Holomorfas
Resumo
O espa¸co de folhea¸c˜oes holomorfas de grau d em Pn forma um espa¸co projetivo PN,
onde N = (n+ 1) d+nn− d−1+n n−1. Em este trabalho estudamos subvariedades de PN
que correspondem a folhea¸c˜oes com comportamentos n˜ao gen´ericos.
Na primeira parte do trabalho encontramos f´ormulas para o grau e dimens˜ao do espa¸co de folhea¸c˜oes de grau d em P2 com singularidades degeneradas. Espec´ıficamente,
estu-damos folhea¸c˜oes com singularidades de ordem k ≥ 2, folhea¸c˜oes com singularidades dicr´ıticas e folhea¸c˜oes com singularidades dicr´ıticas tais que a folhea¸c˜ao induzida no blowup na singularidade tem alguma folha com contato m´aximo com o divisor exepcional no blowup.
Na segunda (e principal) parte do trabalho estudamos outro tipo de comportamento n˜ao gen´erico: admitir subvariedade alg´ebrica invariante. Presentamos f´ormulas para a dimens˜ao e o grau do espa¸co das folhea¸c˜oes de grau d em Pn que tem subespa¸co linear
invariante de quaisquer dimens˜ao. Presentamos f´ormulas para a dimens˜ao e o grau do espa¸co das folhea¸c˜oes de grau d em P2 (resp. P3) que admitem cˆonica invariante, e para
folhea¸c˜oes em P3 que admitem superf´ıcie qu´adrica invariante.
Enumerative Aspects of Holomorphic Foliations
Abstract
The space of holomorphic foliations of dimension one and degree d in Pn form a
projective space PN, with N = (n+ 1) d+n n
− d−1+n n−1. We study subvarieties ofPN
corresponding to foliations having non-generic behavior.
In the first part of the work we find formulas for the degree and the dimension of the spaces of foliations of degree d in P2 displaying degenerated singularities. Specifically,
we study foliations having a singularity of order k ≥ 2, foliations having a dicritical singularity, and foliations having a dicritical singularity such that the foliation induced on the blowup at this singularity has a leave of maximal order of contact with the exceptional line of the blowup.
In the second (and principal) part of the work we study another non-generic behavior: to have an invariant algebraic subvariety. We find formulas for the degree and dimension of the space of foliations of degree d in Pn having an invariant linear subspace of any
dimension. We find formulas for the degree and dimension of the space of foliations of degree d in P2 (resp. P3) having an invariant conic, and for foliations in P3 having an
invariant quadric surface.
Contents
Introduction 3
Chapter 1. Preliminaries 5
1. Dimension one foliations 5
2. Codimension one foliations 8
3. Vector fields versus forms in P2 11
4. Jet bundles 12
5. Order of contact of two curves. 13
6. About the zeros of the adjoint of a section 13
7. Lemmas 14
8. Blowup 15
9. Quadrics 16
10. Local algebra 19
11. Bott’s Formula 20
Chapter 2. Foliations with degenerate singularities 23
1. Singularities of given order 23
2. Dicritical singularities and behavior under blowup 31
Chapter 3. Foliations with invariant algebraic subvarieties 43
1. Foliations with invariant linear subspaces 43
2. Foliations with invariant conic 52
3. Foliations in P3 with invariant quadric 76
Appendix A. 89
1. Foliations with degenerate singularities 89
2. Foliations with invariant algebraic subvarieties 92
Bibliography 109
Introduction
In this work we apply techniques of intersection theory to the global study of holo-morphic foliations in projective spaces.
The last few years have witnessed an important development of the study of holo-morphic foliations. Works of Jouanolou, [22],[23] Cerveau, [5],[6] Lins-Neto, [25],[26], Pereira [7], [8],[29], Cukierman [9],[10],[11], Calvo-Andrade [2],[3] G´omez-Mont [15], [16],[17],[18] and others have focused on global aspects, clarifying questions regarding the (non–)existence of algebraic leaves, description of components for the spaces of folia-tions of codimension ≥1, etc.
Our point of view follows the line of classical enumerative geometry, where are con-sidered questions such as: How many plane curves pass through an appropriate number of points in general position? How many space curves varying in a given family are inci-dent to a determined number of lines in general position? Find the degree of the space of planes curves having a singularity of given order, or compute the degree of the variety of hypersurfaces containing linear subspaces, or conics, or twisted cubics, and so on. . .
In this work we consider similar questions for holomorphic foliations.
Concerning singularities, we compute, in Chapter 2, the dimension and degree of the subvariety of foliations in P2 having singularities of given order, dicritical singularities,
and maximal contact with the exceptional divisor of the blow-up of P2 at a dicritical
singularity.
The rest of the work is dedicated to the study of the spaces of foliations having invariant algebraic subvarieties, first linear subspaces, and then quadrics.
The general philosophy is to find a suitable complete parameter space for the objects we want to study (i.e. foliations satisfying some of the above conditions), enabling us to compute dimension and degree using techniques of intersection theory. In the case of foliations with degenerate singularities (resp. with some invariant linear subspace) the construction of the parameter space is very explicit. In fact, in the case of singularities, we describe them as the projection over the space of foliations of natural projective bundles over P2. For invariant subspaces, we also get a projective bundle over a Grassmannian.
Actually, these bundles are projectivizations of vector bundles the characteristic classes of which we are able to determine.
The case of foliations having an invariant quadric turns out to be subtler and hints at the difficulties to handle degree higher than one. We make and do invoking the classical complete quadrics. In short, we blowup the projective spaces parametrizing quadrics in
4 INTRODUCTION
P2 or P3. In this way we find a compactification of the space of foliations having an
invariant quadric with enough information to compute its degree. Next we describe briefly the contents of each chapter.
Chapter 1 contains the basic notions of foliations of dimension one, as well as of codimension one in projective spaces. We give references to the literature and other definitions and results that we will need in the text.
In Chapter 2 we find formulas for the dimension and degree of the space of foliations of degree d inP2 that have degenerated singularities. In the first section we find parameter
spaces for foliations having a singularity of given order k ≥ 2. In the second section we study foliations with dicritical singularities of order k, and foliations whose strict transform by the blowup of P2 in a dicritical singularity has leaves of maximal contact
with the exceptional divisor.
Chapter 3 is dedicated to the study of foliations having invariant algebraic subvarieties. In the first section of this chapter we find parameter spaces for the variety of foliations having an invariant linear subspace of given dimension; using this description we obtain formulas for its dimension and degree.
In the second and third sections we find a compactification of the space of foliations having a smooth invariant conic (in P2 and inP3) and a smooth invariant quadric in P3.
With this description we find formulas for its dimension and degree.
The last Chapter is an appendix, where we include the scripts inmapleandsingular
that we use in the work.
Natural generalizations of the material covered here include the imposition of flags of invariant subvarieties. For instance, the general foliation in P3 that leaves invariant some
plane does not need to leave any invariant curve therein. So it makes sense to ask for the degree of the space of foliations that leave invariant, say a flag plane ⊃ line, or plane⊃
CHAPTER 1
Preliminaries
In this chapter we introduce the basic notions of foliations of dimension one and codi-mension one in Pn. We review the definitions of degree, singularity, order of a singularity
and invariant subvarieties, in the way that will be used in the text. We also present a few other definitions and results that we will need, such as the notion of jet bundles associated to a vector bundle, scheme of zeros of a section, blowups, complete quadrics, Bott’s formula, including references to the literature. Finally we prove some results that we are going to use but we were not able to find a precise reference for.
Notation 1.1. Throughout this workSdwill denote the vector space of homogeneous
polynomials of degree d in (n + 1)-coordinates Z0, . . . , Zn. We have Sd = SymdCˇn+1;
dimSd= d+nn
. We will identify
S1 = ˇCn+1 ≃Ω0Cn+1
the vector space with basis
{dZ0, . . . , dZn}.
Similarly, we will identify
S1∨ =Cn+1 ≃T 0Cn+1
the vector space with basis ∂ ∂Z0
, . . . , ∂ ∂Zn
.
1. Dimension one foliations
Definition 1.2. Dimension one foliation.
A dimension one foliation in Pn is a global section ofTPn⊗O
Pn(d−1)for somed≥0
modulo non-zero complex multiples.
Let us denote
(1) V1,n,d =H0(Pn, TPn⊗OPn(d−1)) and
F(1, n, d) = P(V1,n,d). Then a dimension one foliation is an elementX ∈F(1, n, d).
Recall the Euler sequence for the tangent bundle of Pn:
6 1. PRELIMINARIES
Tensoring by OPn(d−1) we obtain
0→OPn(d−1)→OPn(d)×Cn+1 →TPn(d−1)→0. Taking global sections in the last sequence and using that H1(Pn,O
Pn(d−1)) = 0 ([21, Chapter III, Theorem 5.1]) we obtain the following exact sequence:
0→Sd−1 →Sd⊗S1∨ →H0(Pn, TPn(d−1))→0.
From this we deduce that a foliation is given in homogeneous coordinates by a vector field
X=F0
∂ ∂Z0
+· · ·+Fn
∂ ∂Zn
where Fi are homogeneous polynomials of degree d, modulo homogeneous polynomial
multiples of the radial vector field R :=Z0
∂ ∂Z0
+· · ·+Zn
∂ ∂Zn
.
We will denote byX an element in Sd⊗S1∨, and X =X modulo Sd−1·R.
The Euler sequence give us:
(3) V1,n,d ≃
Sd⊗S1∨
Sd−1·R
From this it is clear that
N1,n,d := dimV1,n,d−1 = (n+ 1)
d+n n
−
d−1 +n n
−1 and
F(1, n, d) =PN1,n,d. Definition 1.3. Degree of a dimension one foliation.
If X ∈ F(1, n, d) we say that the degree of X is d.
Geometrically, for a generic hyperplaneH inPn, define T(X, H) the set of tangencies
ofX withH. For a genericH, it can be seen thatT(X, H) has codimension one inH and the degree ofX is the degree ofT(X, H) ([27, Chapter II §3]). In fact, take a hyperplane defined by the equation
H :=a0Z0+. . . anZn = 0
then T(X, H) is given in H by
X(H) :=a0F0+· · ·+anFn = 0.
ForH generic, the polynomial X(H) is not identically zero and has degreed.
Observe that in P2 the set of tangencies of a degree d vector field X with a generic
line is finite and consists of d points.
Next, we deduce the local expression of a vector field, using the Euler sequence. If
X =F0
∂ ∂Z0
+· · ·+Fn
1. DIMENSION ONE FOLIATIONS 7
is a degreed vector field, then in
Uj :={[Z0 :· · ·:Zn]|Zj = 06 }={(z0, . . . ,zˆj, . . . , zn)∈Cn}
we have the following local expression forX:
(4) XUj =
n X i=1 ai ∂ ∂zi = n X i=1
(fi−zifj)
∂ ∂zi = n X i=1 fi ∂ ∂zi
−fj n X i=1 zi ∂ ∂zi
wherefi is the dehomogenization ofFi with respect to Zj.
The notation Uj for the canonical open set Zj 6= 0 will be used in all the text.
Definition 1.4. Singularity of a vector field.
We say that p∈Pn is a singularity of X if p is a zero of the section X :OPn →TPn(d−1).
Using the Euler sequence we can compute the number of singularities of a generic vector field of degreed. The fact thatX is generic implies that it has isolated singularities (see [22]), then
#Z(X) =cn(TPn(d−1)).
This can be computed using that c(TPn(d − 1))c(O
Pn(d −1)) = c(OPn(d) ⊗Cn+1) = c(OPn(d))n+1.
In this way we obtain that a dimension one foliation of degree d inPn has
dn+dn−1+· · ·+d+ 1
singularities (counting multiplicities).
Explicitly, using the local expression in (4), we can obtain the singularities of X in Uj as the common zeros ofa1, . . . , an. Alternatively, using homogeneous coordinates, the
singularities of X are given by the ideal of 2 ×2−minors, ZiFj −ZjFi of the matrix Z0 ... Zn
F0 ... Fn
, cf. [13].
Definition 1.5. Order of a singularity.
Let p be a singularity of a vector field X. Suppose that p∈Uj and
XUj =
n X i=1 ai ∂ ∂zi
is the local expression ofX in Uj. Then the order (sometimes named algebraic
multiplic-ity) of the singularityp is
νp(X) = min{orderp(ai)|i= 1. . . , n}.
It can be easily checked that this is independent of the choice of the open Uj.
As usual, the order orderp(a) of a polynomial a at a point p means the order of
vanishing: min{s |∂sa/∂z
I(p)6= 0,|I|=s}.
8 1. PRELIMINARIES
Let Z ⊂Pn be an irreducible hypersurface defined by a homogeneous polynomial G of
degree k, and X be a vector field of degree d. We say that Z is invariant by X if
X(p)∈TpZ
for all p∈Z\(Sing(Z)∪Sing(X)).
If Z is reducible, we say that it is invariant by X if and only if each irreducible component of Z is invariant by X.
Observe that, ifG is irreducible, the above condition is equivalent to the existence of a degree d−1 homogeneuos polynomial H such that
dG(X) = X(G) =GH. This is a consequence of the Hilbert’s Nullstellensatz.
Also note that this condition does not depend on the representative of X. In fact, if X = Y +hR ∈ Sd⊗S1∨ then X(G) = Y(G) +hR(G), but R(G) = kG by the Euler
relation, therefore X(G)∈ hGi if and only if Y(G)∈ hGi. Moreover, if
G=Gr1
1 . . . Grnn
is a decomposition of G into irreducible factors we have that G is invariant by X if and only if Gi is invariant by X for all i= 1, . . . , n.
Definition 1.7. Invariant algebraic subvariety.
If Z ⊂Pn is an algebraic subvariety defined by the ideal I
Z :=hG1, . . . , Gri and X is
a vector field, we say that Z in invariant by X if
X(p)∈TpZ
for all p∈Z\(Sing(Z)∪Sing(X)).
IfIZ is saturated, this condition is equivalent to
dGi(X) = X(Gi)∈IZ
for all i= 1, . . . , r. The hypothesis of the ideal to be saturated is necessary, see [13] p. 5.
2. Codimension one foliations
In this section we define codimension one foliations in Pn (i.e. foliations defined by
integrable one forms in Pn). However, in the text we only deal with codimension one
foliations in P2, so we discuss this case and the correspondence between vector fields and
forms in P2. For further reading see [26]. Definition 1.8. Projective one forms.
A projective one form of degree d inPn is given by a global section of Ω
Pn⊗OPn(d+ 2),
2. CODIMENSION ONE FOLIATIONS 9
As in the case of vector fields in Pn we will deduce an expression in homogeneous
coordinates for a form inH0(Pn,Ω
Pn ⊗OPn(d+ 2)).
Tensoring the (dual of the) Euler sequence (2) by OPn(d+ 2) we obtain (5) 0→ΩPn(d+ 2) →OPn(d+ 1)⊗S1 →OPn(d+ 2)→0
Taking global sections, and using that H1(Pn,Ω
Pn(d+ 2)) = 0 we obtain the following exact sequence:
0→H0(Pn,Ω
Pn(d+ 2))−→Sd+1⊗S1 −→ιR Sd+2 →0 whereιR(PAidZi) = PAiZi is the contraction by the radial vector field.
Then a one form ω ∈H0(Pn,Ω
Pn(d+ 2)) can be written in homogeneous coordinates as
ω =A0dZ0+· · ·+AndZn
where the Ai’s are homogeneous polynomials of degreed+ 1 satisfying
A0Z0+· · ·+AnZn = 0.
Notice that a one form induces a distribution of codimension one subspaces given by p7→Kerωp. However, this distribution is not necessarily integrable.
The condition for this distribution to be integrable is the following:
ω∧dω = 0 For a geometric interpretation see [26] or [4].
Definition 1.9. Codimension one foliations.
A codimension one foliation is an integrable projective one form modulo non zero complex multiples.
Denote
Vn−1,n,d =H0(Pn,ΩPn(d+ 2)). Then a codimension one foliation is given by an element ω of
F(n−1, n, d) :=P(Vn−1,n,d) such thatω∧dω= 0.
Next we deduce the local expression of a projective one form.
LetUj denote the affine open setZj 6= 0, with coordinates (z0, . . . ,zˆj, . . . , zn) then the
local expression of a one form
ω=A0dZ0+A1dZ1+· · ·+AndZn
is
ωUj =a0dz0+· · ·+ [
ajdzj +· · ·+andzn
10 1. PRELIMINARIES
Definition 1.10. Degree of a codimension one foliation.
We say that a foliation given by a one form in F(n−1, n, d) has degree d.
Geometrically, if ω ∈F(n−1, n, d), d is the number of tangencies of the distribution induced by ω with a generic line in Pn. For example, suppose that ω is given in U
0 by
a1dz1+· · ·+andzn
and take the parametrized line l = (t,0, . . . ,0). Then the tangencies of ω with l are given by the zeros of ω|l =a1(t,0, . . . ,0)dt. In principle, a1 has degree ≤ d+ 1, but the
condition of contraction by the radial vector field gives usa0(t,0, . . . ,0) =ta1(t,0, . . . ,0),
so a1(t,0, . . . ,0) has degree ≤ d, and the number of points of tangencies is d. For more
detailed discussion see [26, Proposition 1.2.1 p. 21].
Definition 1.11. Singularities of a one form.
The singularities of a projective one form ω are the zeros of the section ω : OPn → ΩPn(d+ 2).
Locally, if we write ω = a1dz1 +· · ·+andzn the set of singularities of ω is the set of
common zeros of a1, . . . , an.
As in the case of vector fields, a generic one form has isolated singularities and using the sequence (5) we can compute the number of singularities of a generic one form as cn(ΩPn(d+ 2)).
This number is given by
(d+ 1)n−(d+ 1)n−1+· · ·+ (−1)n−i(d+ 1)i+· · ·+ (−1)n. See [22, p. 87].
Definition 1.12. Order of a singularity.
Let p be a singularity of ω. Suppose that p∈U0 and write
ω =a1dz1+· · ·+andzn
for the local expression of ω in U0. Then the order of the singularity p is
νp(ω) = min{orderp(ai)|i= 1. . . , n}.
It can be easily checked that this is independent of the choice of the open Uj.
In what follows we restrict ourselves to the case n = 2.
Definition 1.13. Let ω∈H0(P2,Ω(d+ 2)) and suppose that p∈P2 is a singularity
of ω of order k. If
ωp =akdx+bkdy+h.o.t.
is the local expression ofω, we say thatpisdicritical of order k if akx+bky ≡0 (see [27,
p. 47]).
3. VECTOR FIELDS VERSUS FORMS IN P2 11
Observe that the dicriticity condition is equivalent to ωp =f(x, y)(ydx−xdy) +h.o.t
for some homogeneous polynomial f of degree k−1.
Definition 1.14. Invariant hypersurface.
Let Z ⊂P2 be an irreducible hypersurface defined by a homogeneous polynomial G of
degree k, and F a foliation defined by a one form ω of degree d+ 1. We say that Z is invariant by F if
TpZ ⊂kerωp
for all p∈Z \(Sing(Z)∪Sing(ω)).
If Z is reducible, we say that it is invariant by F if and only if each irreducible component of Z is invariant by F.
If G is irreducible, the above condition is equivalent to the existence of a two-form θ
of degreed such that
dG∧ω=Gθ.
See [22, p. 99].
Definition 1.15. Points of tangency with a hypersurface.
In the definition above, if Z is not invariant, the two-form dG∧ω is not identically zero in Z. The zeros of this form in Z \(Sing(Z)∪Sing(ω)) are the tangencies of F
with Z.
3. Vector fields versus forms in P2
In what follows we study one forms in P2 and their relation with vector fields.
Remark that in P2 a one form is automatically integrable i.e. if
ω =A0dZ0+A1dZ1+A2dZ2
then
ιR(ω) = 0 =⇒ ω∧dω = 0.
Hence a foliation of degree d inP2 can be given by a vector field
X ∈H0(P2, TP2(d−1))
or by a one form
ω ∈H0(P2,Ω
P2(d+ 2)).
Remark 1.16. The reason behind this correspondence is the following. Recall that if
F is a locally free sheaf of rank 2 we have a natural isomorphism
F ≃F∨ ⊗ ∧2F.
12 1. PRELIMINARIES
Using this result for F = ΩP2 we obtain
ΩP2 ≃TP2⊗O(−3). Thus ΩP2(d+ 2) ≃TP2(d−1).
If a foliation inP2 is given by a vector field
X =F0
∂ ∂Z0
+F1
∂ ∂Z1
+F2
∂ ∂Z2
and by a one form
ω=A0dZ0+A1dZ1 +A2dZ2
we can obtain the Fi′s in terms of the A′is and vice versa as follows. Given X the coefficients of ω are expressed by
A0 =Z2F1−Z1F2,
A1 =Z0F2−Z2F0,
A2 =Z1F0−Z0F1.
Given ω, the coefficients of the vector field X can be obtained from dω= (d+ 2)(F0dZ1∧dZ2+F1dZ2∧dZ0+F2dZ0∧dZ1)
This is a consequence of the acyclicity of the Koszul complex associated to the regular sequence {Z0, Z1, Z2}. See [22, §1.5].
Remark 1.17. Locally this correspondence means that if
ωU0 =a1dz1+a2dz2
then a1 =f2−z2f0 and a2 =−(f1−z1f0). Then using (4) we see that
XU0 =−a2 ∂ ∂z1
+a1
∂ ∂z2
.
This corresponds to the intuitive idea that the vector field and the form defining a foliation must be orthogonal to each other.
4. Jet bundles
In this section we recall the notion of jet bundles associated to a vector bundle. A basic reference for this concept is [19, 16.7]. Here we state without proofs the results that we will need in the text.
Let E be a vector bundle over a smooth projective variety X. For n ≥ 0 the n-jet bundle associated toE, denotedPn(E), is a vector bundle over X whose fiber overx∈X
is given by
Pn(E)
x = (OX/mnx+1)⊗ Ex
6. ABOUT THE ZEROS OF THE ADJOINT OF A SECTION 13
For each n ≥0 there exist exact sequences:
(6) 0→Symn+1ΩX ⊗ E → Pn+1(E)→ Pn(E)→0.
As an example take n= 0. Then we have
(7) 0→ΩX ⊗ E → P1(E)→ E →0.
Consider the evaluation map
ev:X×H0(X,E)→ E
given by ev(x, s) = (x, s(x)).
Suppose that we are in a neighborhood of 0 ∈X and that x= (x1, . . . , xn) are local
coordinates of X.
The map ev lifts to a map
ev1 :X×H0(X,E)→ P1(E).
On the fiber of 0 we haveev1(0, s) = (0, s(0) +J0s·x), where J0s is the Jacobian of s at
0. If s(0) = 0 then
ev1(s) = J0s·x∈ΩX,0 ⊗ E0 ≃m/m2 ⊗ E0
i.e. we retrieve the differential of the section.
In general ev lifts to a map evn :X×H0(X,E)→ Pn(E) given by
evn(x, s) = (x, sn(x))
wheresn(x) is the Taylor expansion of s truncated in order n+ 1.
5. Order of contact of two curves.
Definition 1.18. Consider a locally parametrized curve Γ ⊂ C2 given by γ(t), and
an algebraic curve defined by f(x, y) = 0. In a regular point γ(t) of Γ define
i(t) = f(γ(t)).
Clearly, if t0 is a zero of i then γ(t0) is a point of intersection of the curves.
We say that the curves have point-contact of order n at γ(t0) if the n-th derivative of
i(t) is the first non-vanishing derivative of i in t0.
6. About the zeros of the adjoint of a section
Here we state a result that we will need in Chapter 2. For the proofs see [1, p. 16].
Proposition 1.19. Letp:Y →S be a morphism of schemes, F a vector bundle over
Y and s:OY →F a section. Then there exists a section s♭ :OS →p
∗F such that
Z(s♭) ={x∈S |s(y) = 0∀y ∈p−1(x)}.
14 1. PRELIMINARIES
Definition 1.20. Let p : Y → S be a morphism of schemes and u : A → B a
OY-homomorphism of OY-modules.
A closed subscheme Z of S is called the scheme of zeros of u if Z has the following universal property: a map g : T → S factors through it if and only if g∗
Y(u) = 0. If it
exists, the schemes of zeros in denoted ZS(u).
HeregY :T ×SY →Y is the map induced by g and gY∗(u) :gY∗A→g∗YB is given by
g∗Y(u)((t, y), a) = ((t, y), u(y, a)).
Next, we analyze the case in which u : OY → F is a section of a coherent sheaf F over Y. Let us consider, for each x ∈ S the inclusion map g : x → S. Then by definition x∈ZS(u) if and only ifg∗Y(u) = 0, where gY∗(u) is the section of gY∗F given by
gY∗(u)(x, y) = ((x, y), u(y)), for ally∈p−1(x). Therefore,x∈ZS(u) if and only ifu(y) = 0
for all y∈p−1(x). Then ifZS(u) exists, it is equal to {x∈S |s(y) = 0∀y∈p−1(x)}.
In [1] is proved the following result.
Proposition 1.21. Let p:Y →S be as in the above definition and let u: p∗E → F
be an OY-homomorphism. Suppose that E is quasi-coherent and that p∗F is locally free
and its formation commutes with base change. Then the scheme of zeros ZS(u)exists and
is equal to the scheme of zeros of u♭:E →p
∗F.
Again, analyzing the case that u : OY → F is a section of F we obtain that ZS(u)
exists and is equal to the scheme of zeros of u♭ : OS → p
∗F, provided F is well-behaved,
in the sense that it satisfies the hypotheses of Prop.1.21 above. 7. Lemmas
The goal of this section is to prove two lemmas that we are going to use in the sequel. In fact, we will compute the degrees of certain subvarieties of PN which arise as image of
projective subbundles of a trivial bundle X×PN. The second Lemma is more technical
and very usefull to prove one of the hypotheses of the first Lemma.
Lemma 1.22. Let V be a vector space of dimension N + 1. Let K be a sub-bundle of
the trivial bundle X×V over a variety X of dimensionn. Consider P(K) the projective
bundle associated to K, and PN =P(V). Then we have the following diagram:
P(K)
q1
}
}
zzzz zzzz
q2
"
"
E E E E E E E E
X PN
Let M ⊆PN denote the image of q
2. Suppose that q2 is generically injective. Then
degM =
Z
sn(K)∩[X].
8. BLOWUP 15
Then we have degM =
Z
Hν ∩[M] = Z
Hν ∩q
2∗[P(K)] = Z
q2∗Hν ∩[P(K)] = Z
e
Hν∩[P(K)]
whereν = dim(P(K)) = dimM, H =c1OPN(1), He =c1(OK(1)). If r= rk(K), then dim(P(K)) =r−1 +n. Hence
Z e
Hν ∩[P(K)] =
Z
q1∗(Heν ∩q∗1[X]) = Z
sn(K)∩[X]
by definition of the Segre class.
Lemma 1.23. In the situation of the lemma above, in order to prove that q2 is
generi-cally one to one, it suffices to find a point [v]∈M such that the fiber q2−1([v]) consists of
one reduced point.
Proof. If we prove the existence of such point [v], by the theorem on the dimension
of fibers (see [30, Chapter I§6.3]) there exists an open setU inM such that the fiber over each point in U has dimension zero, (U 6=∅ because [v]∈U). Therefore q2 is generically
finite.
Furthermore, by ([12, Chapter II §16.7]) there exists an open set U′ ⊂M where the
cardinality of the fibers is minimal. But the fiber over y = [v] has one point x, if that point is a reduced point.
8. Blowup
In this section we present without proofs the basic facts about the blowup of a scheme along a subscheme that we are going to use in the text. The reference for this subject is [14, Appendix B.6].
Let X be a closed subscheme of a scheme Y, defined by an ideal sheaf J. Then the blowup of Y along X, denoted Ye is defined by
e
Y := Proj(M
n≥0
Jn).
Denote by π : Ye → Y the projection and set E := π−1(X). Suppose that the embedding ofX in Y is regular of codimension d. Then we have
E =P(N)
with projectionη:E →X, whereN =NXY, stands for the normal bundle. Moreover,
(8) NEYe =OYe(E)|E =ON(−1).
16 1. PRELIMINARIES
We have the following exact sequence
0→T E →TYe|E →NEYe →0.
Therefore
(9) TqYe =TqE⊕ON(−1)q
though not canonically. However, ifq∈Y happens to be a fixed point of someC⋆−action
that leavesX invariant, the above decomposition is unique. Ifq = (x,[n]), withx∈X and n∈Nx, then
(10) ON(−1)q =C·n.
In order to computeTqE, we observe that
(11) TqE =TxX⊕T[n]P(Nx).
But T[n]P(Nx) is the fiber over q of the relative tangent bundle of P(N) over X, which is
defined by the following (Euler) exact sequence (see [14, B.5.8.]): 0→OP(N) →η∗N ⊗OP(N)(1)→TP(N)/X →0.
Moreover, from this we deduce that TP(N)/X = Hom(OP(N)(−1),Q) where Q is the
uni-versal quotient bundle of P(N). Thus
(12) T[n]P(Nx) = Hom(C·n,
Nx
C·n). Putting together (9), (10),(11) and (12) we obtain (13) TqYe =TxX⊕Hom(C·n,
Nx
C·n)⊕C·n.
As a final remark observe that asπ is an isomorphism fromYe\E ontoY \X we have, for a pointp∈Ye \E, that
TpYe =Tπ(p)Y.
9. Quadrics
In this section we review the results about quadrics that we will need in the text. A reference for this topic is [33].
9.1. Conics in P2. Let F denote the vector space C3, then P2 = P(F). A conic
is given by a nonzero symmetric map u : F → F∨ modulo non-zero multiples, i.e. an element of P(Sym2(F∨)) = P5.
The rank of a conic is by definition the rank of the mapu. It defines two distinguished subvarieties in P(Sym2(F∨)). The first one is the locus of double lines, corresponding to
the maps with rku = 1. We denote it by V1. The locus of singular conics (the maps
9. QUADRICS 17
det(u) = 0 and V1 =V is the Veronese surface, given by the image of
ν2 :P(F∨)→P(Sym2F∨)
whereν2([a0 :a1 :a2]) = [a20 : 2a0a1 :· · ·:a22] i.e. ν2 sends a lineL to L2.
Next we compute the tangent and normal spaces to the Veronese variety inP5. Suppose
that the double line we are considering isZ2
0, then a vectorv = (a1, a2, . . . , a5) is in TZ2 0V if and only if
Z02+ε(a1Z0Z1+a2Z0Z2+· · ·+a5Z22)∈ V(C[ε])
i.e. if the matrix representing this conic has all 2×2–minors equal to zero over the ring C[ε], ε2 = 0. This matrix reduces (after some elementary operations) to
1 0 0 0 εa3 εa4
0 εa4 εa5
.
So it is clear thatA has rank one if and only if a3 =a4 =a5 = 0. It follows that
TZ2
0V =C·Z
2∨
0 ⊗ hZ0Z1, Z0Z2iC⊂TZ2 0P
5
(14)
Consequently the normal to V inP5 is:
NZ2
0 =C·Z
2∨
0 ⊗ hZ12, Z1Z2, Z22iC.
(15)
In fact, in [33, Proposition 4.4.] is proved that if we consider the Grassmannian of lines inP2 with tautological sequence:
T2 →G×F → Q2
whereT2 has rank 2, then
NVP5 =OP5(1)⊗Sym2(T2∨). This may clarify the description (15).
The Gauss map associated to a conic is the restriction to the conic of the map P(F)→ P(F∨) induced by the linear map u. If a conic C is smooth, this map is an isomorphism
and the dual C∗ is again a smooth conic in ˇP2. But there is no defined envelope of
lines tangent to a singular conic. In order to produce a well defined envelope for every conic, H. Schubert ([31]) introduced the variety of “complete conics”. This variety is a compactification of the variety of smooth conics, different from P5. Let us explain how
this compactification is obtained. Consider the rational map
e1 :P5 =P(Sym2F∨)99KPˇ5 =P(Sym2∧2F∨)
given by e1(u) =∧2u.
The variety of complete conics is the blowup B⊂P5×Pˇ5 ofP5 alongV. In [33, p. 210]
it is proved thatB is embedded inP5×P(Sym
2∧2F∨) and thatB= Graphe1, closure of
18 1. PRELIMINARIES
9.2. Conics in P3. Consider the Grassmannian G = G(2,3) of projective 2–planes in P3. It is endowed with the tautological sequence of vector bundles
0→ T →G×C4 → Q →0
where rk(T) = 3. Pick a plane π ∈G. A conic lying in π is then given by a symmetric map u:Tπ → Tπ∨. Let us defineE := Sym2(T∨). Clearly, the P5−bundle
p:P(E)→G parametrizes the family of conics in P3.
As in the previous section, the rank of a conic defines two distinguished subvarieties in P(E), namely, the locus of double lines (the maps with rku= 1) that we denote V1(T∨)
and the locus of singular conics (the maps with rku≤2) that we denote V2(T∨).
In [33, Proposition 3.2] is proved thatV1(T∨) is the imageV of the Veronese embedding
ν2 :P(T∨)→P(E).
Consider the tautological sequence over the Grassmannian of lines in T:
T2 →G(1,T)× T → Q2.
In [33, Proposition 4.4.] it is shown that
(16) NVP(E) =OE(1)⊗Sym2(T2∨).
As in the case of conics in P2 there exists a rational map e
1, sending a conic to the
envelope of its tangent lines:
e1 :P(E)99KP(Sym2∧2T∨)
given by e1(u) = ∧2u. Its locus of indeterminacy isV.
Denote byBthe blowup ofP(E) alongV. In [33, p.210] it is proved thatBis embedded in P(E)×GP(Sym2∧2T∨) and that B= Graphe1.
9.3. Quadrics in P3. LetF denote the vector spaceC4, so P3 =P(F). A quadric is
given by a nonzero symmetric mapu:F →F∨ up to scalar multiples, i.e. an element of
P(Sym
2(F∨)) = P9.
We have now three distinguished subvarieties in P(Sym
2(F∨)), namely,
• the locus of double planes (the maps with rku= 1) that we denoteV1,
• the locus of pairs of planes (the maps with rku≤2) that we denote V2 and
• the locus of quadratic cones (the maps with rku≤3) that we denote V3.
It is clear that V1 ⊂V2 ⊂V3.
In [33, Proposition 3.2.] is proved that V1 is the image V of the Veronese embedding
10. LOCAL ALGEBRA 19
Consider the Grassmannian G=G(2,3) with tautological sequence 0→ T3 →G×C4 → Q3 →0
where rk(T) = 3. Then we have
N1 :=NVP9 =OP9(1)⊗Sym2(T3∨).
There exists a rational map e1, sending a quadric to the envelope of its tangent lines,
e1 :P9 =P(Sym2(F∨))99KP(Sym2∧2F∨)
given by e1(u) = ∧2u. Its locus of indeterminacy is V. The envelope of tangent lines is
the intersection ofG(1,3)⊂P5 with the quadric of P5 associated to ∧2u.
Denote by B1 the blowup of P9 along V with map π
1 : B1 → P9. Write E1 for the
exceptional divisor,V1
2 ⊂ B1 for the strict transform of V2. In [33, p.210] is proved that
B1 is embedded in P9 ×P(Sym
2∧2F∨) and that B1 = Graphe1.
The subvariety V1
2 is a desingularization of V2. Consider the rational map ρ :V2 99K
G(1,3) given by sending a pair of planes to its intersection line. Then it is clear that the indeterminacy locus of ρ is V1. Therefore if we blow up V2 along V1 we will obtain a
variety isomorphic toV1
2. Here each element determines a line in P3.
Consider the Grassmannian G=G(1,3) with tautological sequence 0→ T2 →G×C4 → Q2 →0
where rk(T2) = 2.
In [33, 6.6] is given the following formula for the normal bundle of V1
2 inB1:
N :NV1
2B1 =OP9(1)⊗Sym2(T
∨
2 )⊗OB1(−E1)|E1 whereOB1(−E1)|E1 =ON1(1) (see8).
Now, consider the rational map e2, sending a quadric to the envelope of its tangent
planes,
e2 :P9 =P(Sym2(F∨))99KP(Sym2∧3F∨).
It is given bye2(u) =∧3u. The locus of indeterminacy is V2. Define e
e2 :=e2◦π1 :B1 99KP(Sym2∧3F∨).
Then the locus of indeterminacy of ee2 isV21. Denote by B the blowup of B1 along V21. In
[33, Theorem 6.3] is proved that B is embedded in P9 ×P(Sym
2∧2F∨)×P(Sym2∧3F∨)
and that B= Graphee2 = Graphe1×e2.
10. Local algebra
In this section we prove a technical lemma that will be used repeatedly in Chapter 3.
Lemma 1.24. LetRbe a local Noetherian domain, and ϕ:Rn →Rm a homomorphism
20 1. PRELIMINARIES
Proof. Let
A=
a11 ... a1n ... ... ...
am1 ... amn
be the m×n matrix associated to ϕ, i.e. the columns of A generate M. By hypotheses for k = 1, the ideal of coefficients of A is principal:
ha11, . . . , aij, . . . , amni=hfi.
We may assume f 6= 0. Let bij := afij. We may suppose a11 =f. Let M′ be the module
generated by the columns of
B =
1 ... b1n ... ... ...
bm1 ... bmn
.
Equivalently (by elementary operations)M′ is generated by the columns of (1 0
0B′), where B′ =
b22 ... b2m ... ... ...
bm2 ... bmn
.
Applying inductive hypotheses, we have that ImB′ is free. Thus M′ is free. Since R
is a domain we have M=fM′ ≃ M′. Then Mis free.
11. Bott’s Formula
In this section we explain Bott’s equivariant formula in the basic context that we are going to use. A reference for this subject in the general case is [28] and the bibliography therein.
Let X be a smooth complete variety of dimension n, and let T =C⋆ act on X with
isolated fixed points. WriteXT for the set of fixed points. LetE be a T-equivariant fiber
bundle over X of rankr.
If p(c1, . . . , cr) is a weighted homogeneous polynomial of total degree n with rational
coefficients, where degci =i, then
p(c1(E), . . . , cr(E))∩[X]
is a zero cycle in X. Bott’s formula expresses the degree of that zero cycle in terms of data given by the action ofT on the fibers ofE and the tangent bundleT X over the fixed points of the action. Below is an outline.
Letp∈XT be a fixed point. The torusT acts on the fiberEp and (asT is semisimple)
we have a complete decomposition of Ep into eigenspaces, with certain weights ξi ∈Z: Ep =⊕ri=1Eξi
p .
Thus Eξi
p ={v ∈ Ep |t·v =tξiv, t ∈T}.
Set cT
i (Ep) := σi(ξ1, . . . , ξr), where σi denotes the i-th elementary symmetric
polyno-mial:
σ1 = X
ξj, σ2 = X
i<j
ξiξj, . . . , σr =ξ1· · ·ξr.
Set pT(E
11. BOTT’S FORMULA 21
Theorem 1.25. (Bott’s formula) Z
p(c1(E), . . . , cr(E))∩[X] = X
p∈XT
pT(Ep)
cT n(TpX)
.
CHAPTER 2
Foliations with degenerate singularities
In this chapter we study foliations in P2 that have degenerate singularities. The
first type of degeneration we will consider is given by the order of the singularity. If ω∈PN, (N =N
1,2,d) defines a generic foliation of degreedinP2, we expect its singularities
to have all order one; then we study the complementary condition, i.e. we ask for the space of foliations that have some singularity of order≥k. We describe this space as the imageMk by the second projection ofP(Rk)⊂P2×PN, whereRk is a vector bundle over
P2 the characteristic classes of which we are able to determine.
In the second part we study the space Dk ⊂ Mk of foliations that has a dicritical
singularity of orderk. Again, this is a closed condition, and we construct a parameteriza-tion of that space. We find a subvector bundleDk of the trivial bundle V1,2,d (cf. (3, p.6))
overP2, such that the image by the second projection of P(Dk) is D
k. We determine the
characteristic classes ofDk, and with this at hand we can compute the degree ofDk.
If a foliationF has a dicritical singularityp, then the exceptional divisor of the blowup of P2 at p is not invariant by the strict transformF∗ of F. The leaves of F∗ may not be
transverse to the exceptional divisor at some points. In the last part of this chapter we study the space of foliations such thatF∗ has a leaf with maximal order of contact with
the exceptional divisor at some point. We obtain a sufficiently explicit description of this variety enabling us to compute its codimension and degree inPN.
Requiring a leaf of a foliation to be tangent to a line at a given point defines a hyperplane in PN. Thus, finding the degree of the loci C
k ⊂ Dk ⊂Mk can be rephrased
loosely as calculating the number of foliations with a singularity of the chosen type and further tangent to the appropriate number of flags (point, line) inP. It turns out that the degrees ofCk, Dk, Mk are expressed as explicit polynomials ink, d.
1. Singularities of given order In order to simplify the notation we set
V :=V1,2,d =H0(P2,ΩP2(d+ 2)) and N :=N1,2,d = dimV −1.
Fix k ≤d+ 1. In this section we describe a parameter space Mk ⊂PN for the locus
of foliations of given degreed that have some singularity of order ≥k. In fact we obtain a filtration ofPN,
Md+1 ⊂. . . M3 ⊂M2 ⊂PN.
In Proposition (2.5, p.27) we show that the codimension of Mk in PN is
codMk =k(k+ 1)−2.
24 2. FOLIATIONS WITH DEGENERATE SINGULARITIES
Then we ask the following question: Given a (k(k+ 1)−2)−dimensional linear familyFt
of foliations, how many members ofFtwill have some singularity of order≥k? Of course the answer to this question is given by the degree of Mk, and we find in Proposition 2.5
that
deg(Mk) = Z
c2(Pk−1(Ω(d+ 2)))∩[P2]
where Pk−1(Ω(d+ 2)) is the (k−1)−jet bundle associated to Ω(d+ 2).
Recall (Definition 1.12, p.10) that if ω ∈H0(P2,Ω(d+ 2)), the order of a singularity
p∈P2 is
νp(ω) := min{orderp(a), orderp(b)},
where ωp =adx+bdy is a local expression ofω in a neighborhood ofp.
1.1. Order one. Consider the map of fiber bundles over P2,
ev:P2×V →Ω(d+ 2)
given by evaluation, ev(p, ω) = (p, ω(p)).
We claim that ev is surjective. In fact, it is sufficient to prove the surjectivity in the fibers. For this suppose p= [0 : 0 : 1] and let λdx+µdy∈Ωp(d+ 2). Then
ω :=Z2d(Z2λdZ0+Z2µdZ1−(Z0λ+Z1µ)dZ2)∈H0(P2,Ω(d+ 2))
satisfies ω(p) =λdx+µdy.
Remark 2.1. In fact Ω(d+ 2) is generated by global sections (see [22, Lemma 2.3.6,
p. 90]).
Set R:= Ker(ev). Then R is a subbundle of V of rank rkR= dimV −2 = N −1. It fits into the following exact sequence
(17) 0→ R −→P2×V −→Ω(d+ 2)→0.
Definition 2.2. The universal singular set is the projective bundle associated toR,
i.e. the incidence variety:
P(R) ={(p,[ω])|p is singularity of [ω]} ⊂P2 ×PN.
Let us denote byp1, qthe projections ofP(R) in the first and second factor respectively.
We have the diagram
P(R)
p1
|
|
zzzz zzzz
q
"
"
E E E E E E E E
P2 PN
1. SINGULARITIES OF GIVEN ORDER 25
We may compute the cardinality of a generic fiber as follows. Observe thatq∗[P(R)] =
deg(q)[PN]. Write H :=c
1(OPN(1)). We have deg(q) =
Z
HN ∩q∗[P(R)] = Z
q∗(c1(q∗OPN(1))N ∩[P(R)]) =
Z
c1(OR(1))N ∩[P(R)] = Z
p1∗(c1(OR(1))N ∩p∗1[P2])
=
Z
s2(R)∩[P2].
The last equality follows by definition of the Segre class, recalling that rk(R) = N −1. In this way we retrieve the well know number of singularities of a degree d foliation. Indeed, by (17) we have s2(R) =c2(Ω(d+ 2)). From the Euler sequence we find
c2(Ω(d+ 2)) =d2+d+ 1.
1.2. Order two. Now consider the first jet bundleP1(Ω(d+ 2)) of Ω(d+ 2). It fits
into the following exact sequence (see (7, p.13) of Chapter 1):
(18) 0→Ω⊗Ω(d+ 2)→ P1(Ω(d+ 2))→Ω(d+ 2) →0.
Consider the following diagram:
Ω⊗Ω(d+ 2)
P1(Ω(d+ 2))
6
6
r |
R //
J1
5
5 55
V ev // Ω(d+ 2). Notice that we can lift ev to a map
V → P1(Ω(d+ 2)).
The composition of the horizontal maps is zero, so we have an induced map J1 :R →Ω⊗Ω(d+ 2).
It is interpreted in the fibers asJ1(p, ω) = (p,Jpω), i.e., the part of degree 1 of ω.
Lemma 2.3. The map
J1 :R →Ω⊗Ω(d+ 2)
is surjective for all d≥1.
Proof. We shall prove that for allp∈P2 the map on the fibers is surjective. Assume
that p= [0 : 0 : 1], and take coordinates (x, y) in U2. Let m =hx, yi. Pick
26 2. FOLIATIONS WITH DEGENERATE SINGULARITIES
with a1, b1 linear homogeneous polynomials inx, y. Here we use the basis{x, y}of m/m2
for the first factor and the basis {dx, dy} for the second factor Ωp.
Take the following degree 2 form:
ω =Z2a1(Z0, Z1)dZ0+Z2b1(Z0, Z1)dZ1−(Z0a1+Z1b1)dZ2.
Then bω:=Z2d−1ω is a form of degreed+ 1 that has a singularity at p with part of degree
one equal to θ.
1.2.1. Let R2 denote the kernel of J1. Thus R2 is a fiber bundle that fits into the
following exact sequence:
(19) 0→ R2 → R −→Ω⊗Ω(d+ 2)→0.
Recalling (17, p.24), we have
rk(R2) = rkR −4 = dimV −6.
The projective bundle associated to R2 is the incidence variety,
P(R2) = {(p,[ω])|p is a singularity of [ω] withνpω ≥2} ⊂P2×PN.
Denote by p1, q the projections ofP(R2) in the first and second factor respectively.
P(R2)
p1
|
|
yyyy yyyy
y q
#
#
F F F F F F F F
P2 PN
ClearlyM2 :=q(P(R2)) is the subvariety ofPN of forms with some singularity of order
at least two.
In order to find the degree of M2 we have to check thatq is generically injective, and
then compute s2(R2) (cf. Lemma 1.22, p.14).
Lemma 2.4. The morphism q:P(R2)→PN is generically injective.
Proof. For each d ≥ 1 we will exhibit a form ω of degree d+ 1 that has a unique
singularity pwith order two and such thatd(q,[ω])q is injective. This suffices to prove that
q is generically injective (see Lemma 1.23, p.15). We claim that
ω:= (Z0Z2d−1+Z1Z2d−1+Z0d+ (−Z1)d)(−Z1dZ0+Z0dZ1)
has the required properties.
In fact, in the affine chartU2 ={Z2 6= 0} we have
ω= (x+y+xd+ (−y)d)(−ydx+xdy).
It’s clear that (0,0) is a singularity with order two and is easy to see that any other singularity inU2has order one. Indeed, writef(x, y) :=x+y+xd+(−y)d. If (α, β)6= (0,0)
1. SINGULARITIES OF GIVEN ORDER 27
f(α, β) = 0,
∂f
∂x(α, β) = ∂f
∂y(α, β) = 0
and this implies α=β = 0.
In the chart U0 (U1 respectively) the local expression ofω is
ω= (zd−1+yzd−1+ 1 + (−y)d)dy
(ω =−(xzd−1+zd−1+xd+ (−1)d)dx respectively).
In both cases is easy to see that the singularities have order one.
Next we prove that dq is injective at (p,[ω]). Consider a vector ((p1, p2), θ)) ∈
T(p,[ω])P(R2). We claim thatθ = 0 implies p1 =p2 = 0.
In fact, the vector above is the tangent vector to the infinitesimal curve ([εp1 :εp2 : 1], ω+εθ)
inP(R2) if and only if the point is a singularity of order two (working in C[ε]/hε2i).
Writing θ =adx+bdy and using that ε2 = 0 we see that (εp
1, εp2) is a singularity of
ω+εθ if and only if a(0,0) =b(0,0) = 0. In fact, ω((εp1, εp2)) = 0 and εa((εp1, εp2)) =
ε(a(0,0) +εp1ax(0,0) +εp2ay(0,0) +h.o.t) =εa(0,0) = 0 then a(0,0) = 0. The same is
valid for b.
On the other hand, the point has order two if and only ifJω+εJθvanishes at (εp1, εp2).
Now,
Jω+εJθ(εp1, εp2) has the following form:
−ε(p2(1 +d(εp1)d−1) +ax) ∗
∗ ε(p1(1 +d(−εp2)d−1) +by) !
Therefore if θ = 0, we see that the condition for the point to be of order two implies
p1 =p2 = 0 as claimed.
1.2.2. Chern classes of R2. Using sequences (19), (17) and (18) we compute s(R2):
s(R2) =c(Ω(d+ 2))c(Ω⊗Ω(d+ 2)) =c(P1(Ω(d+ 2))).
1.3. Order k≥3. We now generalize the above procedure.
Proposition 2.5. For 2≤k ≤d+ 1, denote by
Mk={[ω]|[ω] has a singularity of order at least k} ⊂PN.
Then we have
codMk =k(k+ 1)−2
and
deg(Mk) = Z
P2
c2(Pk−1(Ω(d+ 2))).
Proof. In general we have, forn ≥1 the following exact sequence (cf.6, p.13):
28 2. FOLIATIONS WITH DEGENERATE SINGULARITIES
As in the case k = 2 we will define inductively a subbundle Rk of V such that P(Rk)⊂P2×PN is the incidence variety
{(p,[ω])|pis singularity of [ω] andνpω≥k}.
Suppose that we have defined Rr for r < k. In order to define Rk observe that we
have a map
ek−2 :Rk−2
Jk−2
−→Symk−2Ω⊗Ω(d+ 2) −→ Pk−2(Ω(d+ 2)).
Consider the following diagram:
Symk−1Ω⊗Ω(d+ 2)
Pk−1(Ω(d+ 2))
4
4
n t |
Rk−1 //
Jk−1
4
4 44
Rk−2
ek−2
/
/ Pk−2(Ω(d+ 2)) We can lift ek−2 to a map
Rk−2 → Pk−1(Ω(d+ 2)).
Using that the composition of the horizontal maps is zero, we obtain an induced map (21) Jk−1 :Rk−1 →Symk−1Ω⊗Ω(d+ 2)
that is interpreted on the fibers as Jk−1(p, ω) = (p,(ωp)k−1) (the part of degree k−1 of
ωp). Here ωp stands for the local expression of ω in a neighborhood of p.
We claim that Jk−1 is surjective. Indeed, we prove that the map on the fibers is
surjective.
Assume that p= [0 : 0 : 1], and take local coordinates (x, y). Consider θ =ak−1⊗dx+bk−1⊗dy∈Symk−1Ωp⊗Ωp = Symk−1(m/m2)⊗Ωp.
Hereak−1, bk−1 are degreek−1 homogeneous polynomials in (x, y). As before we use the
basis {x, y} of m/m2 for the first factor and the basis {dx, dy} for the second factor Ω
p.
Take the following degreek form: ˜
ω =Z2ak−1(Z0, Z1)dZ0+Z2bk−1(Z0, Z1)dZ1−(Z0ak−1+Z1bk−1)dZ2.
Then ˆω := Z2d−k+1ω˜ is a form of degree d+ 1 that has a singularity at p with part of degree k−1 equal to θ.
Set Rk := KerJk−1. Thus, Rk is a fiber bundle of rank
rk(Rk) = rk(Rk−1)−2k,
(♥)
that fits into the following exact sequence:
1. SINGULARITIES OF GIVEN ORDER 29
By construction, the projective bundle associated toRk is the incidence variety P(Rk) ={(p,[ω])|pis singularity of [ω] andνpω≥k} ⊂P2×PN.
As before, let q : P(Rk) → PN denote the projection in the second factor. We
prove that q is generically injective in Lemma 2.6 below. Now, in order to compute deg(Mk) =
R
s2(Rk)∩[P2] we use sequence (22), the case k−1, and (20) to obtain
s(Rk) = c(Pk−2Ω(d+ 2))c(Symk−1Ω⊗Ω(d+ 2)) =c(Pk−1(Ω(d+ 2))).
To compute the codimension ofMk, assuming thatq is finite, and using (♥) we obtain
that cod(Mk) =N −dimP(Rk) = N −(rk(Rk) + 1) =N −(rk(Rk−1)−2k+ 1)
=N −dim(P(Rk−1)) + 2k = cod(Mk−1) + 2k.
From this relation and the fact that cod(M1) = 0 we deduce that
cod(Mk) = k(k+ 1)−2.
It remains to show
Lemma 2.6. For all 3 ≤ k ≤ d+ 1 the projection q : P(Rk) → PN is generically
injective.
Proof. As before we shall find a formω of degreed+ 1 that has a unique singularity
pof order k and such thatd(p,[ω])q is injective.
We claim that the following form has this properties: ω = (Z2d−(k−1)Z0k−1+Z
d−(k−1)
2 Z1k−1+Z0d+Z1d)(−Z1dZ0+Z0dZ1).
Indeed, if we denote F :=Z2d−(k−1)Z0k−1+Z2d−(k−1)Z1k−1+Zd
0 +Z1d, in the chartU2 we
have
ω = (xk−1+yk−1+xd+yd)(−ydx+xdy) =f(−ydx+xdy) =adx+bdy.
Is clear that (0,0) is a singularity of order k. Suppose that (α, β) is another singularity. Then we have
f(α, β) =αk−1+βk−1+αd+βd= 0. On the other hand, if the first jet ofω at that point is zero we have
∂a
∂x(α, β) = −β ∂f
∂x(α, β) =−βα
k−2
(k−1 +dαd−k+1) = 0 ∂a
∂y(α, β) = −β ∂f
∂y(α, β) =−β
k−1(k−1 +dβd−k+1) = 0
∂b
∂x(α, β) = α ∂f
∂x(α, β) = α
k−1(k−1 +dαd−k+1) = 0
∂b
∂y(α, β) = α ∂f
∂y(α, β) = αβ
k−2
30 2. FOLIATIONS WITH DEGENERATE SINGULARITIES
Suppose that α 6= 0. Then ∂f∂x(α, β) = 0, and if (α, β) is a singularity of order two, then
∂2b
∂2x(α, β) = α
∂2f
∂2x(α, β) = α
k−2((k−1)(k−2) +d(d−1)αd−k+1) = 0
i.e.
((k−1)(k−2) +d(d−1)αd−k+1) = 0. This together with
dαd−k+1 =−(k−1)
give us k−2−(d−1) =k−1−d= 0 i.e. k =d+ 1. It is easy to see that if k =d+ 1 the unique singularity of order greater than one is (0,0).
Thereforeα = 0, and similarly β = 0.
On the other charts, for example in U0 the expression of the form is
(zd−k+1(1 +yk−1) + 1 +yd)dy. It is easy to see that the singularities are of order less than two.
It remains to show thatd(p,[ω])qis injective. For this, we consider a vector ((p1, p2), θ))∈
T(p,[ω])P(Rk), and we have to prove that θ= 0 implies p1 =p2 = 0.
The vector above is the tangent vector to a curve ([εp1 : εp2 : 1], ω+εθ) in P(Rk) if
and only if the point is a singularity of order k (working inC[ε]/hε2i).
Suppose that Ji(ω+εθ)(εp1, εp2) = 0 for all i ≤ k−1. It is easy to see that in this
case, if θ = 0, Jk−1(ω+εθ)(εp1, εp2) = 0 implies:
−εp2
∂k−1f
∂k−1x(εp1, εp2) = 0
−εp1
∂k−1f
∂k−1x(εp1, εp2) = 0
But ∂∂kk−1−1fx = ((k −1)! +d(d−1)· · ·(d−k + 2)xd−k+1). Hence the above equations implies
εp2(k−1)! = 0
εp1(k−1)! = 0
i.e. p1 =p2 = 0.
1.3.1. We claim that c2(Pk−1(Ω(d+ 2))) is a polynomial of degree ≤ 2 in d and of
degree ≤6 in k.
In fact, from (20) we derive the following relation for the Chern character ofPk−1(Ω(d+
2)):
2. DICRITICAL SINGULARITIES AND BEHAVIOR UNDER BLOWUP 31
15]), we obtain:
ch(SymkΩ) = (k+ 1)−3
2(k
2+k)h+ (−1
4k+ 3 4k
2+k3)h2
whereh=c1(OP2(1)). Then
ch(Symk−1Ω) +ch(Symk−2ΩP2) +· · ·+ch(Ω) + 1 =b0+b1h+b2h2 where each bi is a polynomial in k if degree ≤i+ 2.
On the other hand, ch(Ω(d+ 2)) =a0+a1h+a2h2, where each ai is a polynomial in
d if degreei.
Summarizing, we have that
ch(Pk−1(Ω(d+ 2))) =d0+d1h+d2h2
wheredi is a polynomial in d if degreei in dand of degree ≤i+ 2 ink. Hence, from the
definition of Chern character,
c2(Pk−1(ΩP2(d+ 2))) = 1 2d
2 1−d2
i.e. a polynomial of degree 2 ind and of degree ≤6 in k.
Using Proposition (2.5, p.27), and the above claim, we may now derive an explicit formula for the degree ofMk inPN: it suffices to do it for the first few values of d, k and
interpolate. We use amaple procedure (see Appendix Section1.1) and obtain Corollary 2.7. The degree of Mk is given by the following formula:
1
2k(k+ 1)
1 4(4k
4−8k3−7k2+ 21k−6)−(2k−3)(k2+k−1)d+ (k2+k−1)d2
.
2. Dicritical singularities and behavior under blowup
Recall that ifω∈H0(P2,Ω(d+2)) andpis a singularity ofω, we say thatpis dicritical
if the local expression ofω is
ωp =akdx+bkdy+h.o.t
with akx+bky = 0 (Definition 1.13, p.10). To have a dicritical singularity will be shown
to be a closed condition inPN. This will be rephrased in a coordinate-free manner.
In the first part of this section we find a parameter space Dk for the forms that have
a dicritical singularity of orderk.
In Proposition (2.9, p.33) we obtain that the codimension of Dk is
k(k+ 2)
and the degree of Dk is given by the coefficient of the degree two part of
c(Pk−1(Ω(d+ 2)))c(Sym
32 2. FOLIATIONS WITH DEGENERATE SINGULARITIES
Next, suppose thatpis a singularity ofωof orderk ≥2 (for the casek = 1 see Remark
2.16, p.41), and let F denote the foliation induced byω. Take an affine chart p∈U ≃C2 and consider
π :Ce2 →C2
the blowup of C2 at p. Then we can define (by pullback) a foliation F∗ in Ce2 (see
[27]). The fact that pis a dicritical singularity implies that the exceptional divisor of the blowup is not invariant by F∗. As we will see, in the dicritical case, the foliation F has
an infinite number of leaves through p, that come (by blow-down) from leaves ofF∗ that
are transverse to the exceptional divisor. On the other hand, there exists a finite number of points of tangency of the leaves ofF∗ with the exceptional divisor. Each of them gives
(by blow-down) leaves of F whose closure has a singularity atp.
2.1. Maximal contact. We study the condition in ω for F∗ to have a leaf with
maximal order of contact with the exceptional divisor. We name this property MCE. In the second part of this section we describe a parameter spaceBk for the foliations
that have some singularity with the MCE property.
In Proposition (2.15, p.40) we obtain the formula for the codimension and for the degree of Bk.
2.2. Dicritical singularities. Next we give an invariant way of expressing the con-dition that a singularity be dicritical.
Suppose thatE is a vector bundle of rank 2. Then for allk ≥1 we have the following exact sequence (e.g., see [12, Appendix 2 A2.6.1.]):
0→ ∧2E ⊗Symk−1E →SymkE ⊗ E Pk
→Symk+1E →0, where the first map is given by
(a∧b⊗c)7→(ac⊗b)−(bc⊗a) and the second by
a⊗b 7→ab.
2.2.1. Dicritical of order one. Recall from the previous section the surjective map (cf.21)
J1 :R →Ω⊗Ω(d+ 2).
We can composeJ1 withP1 : Ω⊗Ω(d+ 2)→Sym2Ω⊗OP2(d+ 2) to obtain the surjective map
T1 :R →Sym2Ω⊗OP2(d+ 2).
This map is interpreted in the fiber of a point p ∈ P2 as follows. Suppose that
