1. Introduction
1.3. Overview of the paper
The proof of Theorem 1.1 consists of two parts: the description of the local strategy for resolution of singularities, given in§4, and the proof that this local strategy can be globalized, which will be explained in§5.
Let us briefly describe the ingredients used in the central result of the paper: the theorem on local resolution of singularities.
For definiteness, we assume here that M=(R3(x,y,z),0), and that the origin is con-tained in a divisor with normal crossingsDwhich is given either by{(x, y, z)∈R3:x=0}
(briefly,{x=0}) or by{xy=0}. We further assume that the reduced vector field χ de-fined in M is tangent to the divisorD, and that the origin is a nonelementary singular point. Finally, we assume that the vertical axis {x=y=0} is not entirely contained in the set NElem of nonelementary points.
Using the logarithmic basis{x∂/∂x, y ∂/∂y, z ∂/∂z}, we can write χ=f x ∂
∂x+gy ∂
∂y+hz ∂
∂z,
h
n
h
n
h
n
v2
v3
(−1,0) (0,−1)
(−1,0) (0,−1)
(−1,0)
(0,−1)
(a) (b) (c)
Figure 1. Regular and nilpotent configurations.
wheref x,gyandhzare germs inR{x, y, z}. TheNewton polyhedron ofχ(with respect to the coordinates (x, y, z)) is the convex polyhedron
N= conv(supp(f, g, h))+R3>0,
where conv(·) denotes the operation of convex closure, supp(f, g, h)⊂Z3 is the set of integer pointsv=(v1, v2, v3)∈Z3 such that the monomialxv1yv2zv3 has a nonzero coef-ficient in the Laurent expansion of either f, g or h; and the “+” operator is the usual Minkowski sum of convex polyhedrons.
The higher vertex of N is the vertex h∈N which is minimal with respect to the lexicographical ordering in R3. By the hypothesis, it follows that this vertex has the formh=(0, h2, h3) for some integersh2, h3∈Z>−1. Moreover, the intersection ofN with the plane{v∈R3:v1=0}is in one of the situations shown in Figure 1.
Referring to Figure 1, the configurations (a) and (b) are called regular and the configuration (c) is callednilpotent. As indicated in the figure, we define themain vertex m=(m1, m2, m3) by m=h in cases (a) and (b), and by m=n in case (c). Now, we consider the intersection
N0=N ∩
v∈R3:v3=m3−12 , and call the polygonN0 thederived polygon (see Figure 2).
The derived polygon has some similarities with thecharacteristic polygonintroduced by Hironaka [H3] in his proof of the resolution of singularities for excellent surfaces.
However, there are some essential differences which will be discussed in§3.3.
Let us denote by m0= m01, m02, m3−12
the minimal vertex of N0 (with respect to the lexicographical ordering) and write the displacement vectorm0−mas 12(∆1,∆2,−1), for some nonzero rational vector ∆=(∆1,∆2)∈Q2.
v1
v2
v3
m m0
1 2
N0
v1
v2
m0
Figure 2. The derived polygon.
Themain invariant for the vector fieldχ (with respect to the coordinates (x, y, z)) is given by the 6-tuple of natural numbers
inv = (h, m2+1, m3,#ι−1, λ∆1, λmax{0,∆2}),
whereλ=(m3+1)!, #ι∈{1,2}is the number of local irreducible components of the divisor at the origin and thevirtual height his the natural number defined by
h=
bm3+1−1/∆2c, ifm2=−1 and ∆1= 0, m3, ifm2= 0 or ∆1>0, wherebαc:=max{n∈Z:n6α} (see Figure 3 for an example).
In §4.5 we shall introduce the fundamental notion of stable coordinates. Roughly speaking, if we start with a system of local coordinates (x, y, z) as above, we obtain a new system of coordinates (˜x,y,˜ z) by an analytic change of coordinates of the form˜
˜
x=x, y˜=y+G(x) and z˜=z+F(x, y), (2) in such a way that the main invariant inv, when computed with respect to the new coordinates (˜x,y,˜ z), has nice analytic properties such as being an upper semicontinuous˜ function. Moreover, we shall see that inv is an intrinsic object attached to the germ of vector fields χ, up to fixing an additional geometric structure on the ambient space called anaxis (see§2.7).
The local strategy of reduction of singularities will be read out of the Newton poly-hedron N and the main invariant inv, provided that these objects are computed with respect to a stable system of coordinates.
v1
v2
v3
m=(0,−1,4)
(0,1,1)
(0,4,−1) (1,−1,1)
0,23,−1
⇒(∆1,∆2)= 0,23 m3− 1
∆2
=5
2 ⇒h=3
inv= 3,0,4,0,0,235!
Figure 3. The Newton polyhedron forχ=x2y ∂/∂x+(z4+xz)∂/∂y+y4∂/∂z.
The notion of stable coordinates is similar to the notions of well-prepared andvery well-prepared systems of coordinates, as defined by Hironaka [H3] in the context of func-tion germs. However, new difficulties appear in the context of vector fields, since the action of the Lie group of coordinate changes given in (2) is much harder to study in this situation. An example of new phenomena is the appearance of the so-called resonant configurations, described in§4.4.
Let us now briefly introduce a second ingredient of our proof: the notion of weighted blowing-up. Given a vector of nonzero natural numbers ω=(ω1, ω2, ω3)∈N3>0, the ω-weighted blowing-up (with respect to the coordinates (x, y, z)) is the proper analytic surjective map given by
Φω:S2×R+−!R3,
((¯x,y,¯ z), τ)¯ 7−!(x, y, z) = (τω1x, τ¯ ω2y, τ¯ ω3z).¯
Similarly, for a weight-vector of the formω=(ω1,0, ω3) withω1 andω3nonzero, we define
Φω:S1×R+×R−!R3,
((¯x,z), τ, y)¯ 7−!(x, y, z) = (τω1x, y, τ¯ ω3z).¯
We define analogously the blowing-ups for weight-vectorsω of the forms (0, ω2, ω3) and (ω1, ω2,0). Notice that the blowing-up center, in these four cases, is given respectively by{x=y=z=0},{x=z=0},{y=z=0}and{x=y=0}.
Suppose now that (x, y, z) is a stable system of coordinates at the origin, and let inv∈N6be the corresponding main invariant. Starting from§4.8, we prove the following result on local resolution of singularities for vector fields: There exists a choice of weight-vector ω for which the strict transform χe of the vector field χ under the ω-weighted blowing-up
Φω:Me−!M
is such that, for each nonelementary point ˜p∈Me∩Φ−1(0), and each choice of a stable system of coordinates (˜x,y,˜ z) with center at ˜˜ p, the corresponding main invariantinv isf such that
invf<lexinv, where<lexis the usual lexicographical ordering inN6.
The second part of the proof of Theorem 1.1 is given in§5. There we show that the local strategy for resolution of singularities described in the previous paragraph can be globalized.
To prove this, the main ingredient is the fact that both the main invariant inv and the choice of a weight-vector ω are independent of the given stable system of coordinates.
Moreover, we shall see in §5.2 that inv is an upper semicontinous function on the set NElem of nonelementary singular points of the vector field.
Based on these facts, the global strategy of resolution is similar to the one presented by Cano in [C2], based on the notion of generic equireducibility and bad points. The main distinction is the fact that our strategy leads to a unique choice of local center and ana priori absence of cycles, due to a conveniently chosen enumeration of the exceptional divisors. Moreover, we can guarantee that the blowing-up centers are always contained in the set of nonelementary points.