Abstract
We prove Local Uniformization for arbitrary excellent hypersurface threefolds of multiplicity smaller than the residue characteristic. This article is part of the authors’ Resolution of Singularities program for arithmetic varieties of dimension three. The proof builds upon Hironaka’s characteristic polyhedron and invariants.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
This article is part of the authors’ program whose purpose is to prove the following conjecture on Resolution of Singularities of threefolds in mixed characteristic. The conjecture is a special case of Grothendieck’s Resolution conjecture for quasi-excellent schemes.
Conjecture 1.1
Let \(C\) be an integral regular excellent curve with function field \(F\). Let \(S/F\) be a reduced algebraic projective surface and \(\mathcal{X}\) be a flat projective \(C\)-scheme with generic fiber \(\mathcal{X}_F=S\). There exists a birational projective \(C\)-morphism \(\pi : \mathcal{Y}\rightarrow \mathcal{X}\) such that
-
(i)
\(\mathcal{Y}\) is everywhere regular.
-
(ii)
\(\pi ^{-1}(\text{ Reg}\mathcal{X})\rightarrow \text{ Reg}\mathcal{X}\) is an isomorphism.
Let us point out that the equicharacteristic techniques designed in [12] extend to the situation described in the above conjecture. In particular, [12] theorem 3.3 extends and reduces Conjecture 1.1 to the following variant:
Conjecture 1.2
Let \(A\) be an excellent DVR with quotient field \(F\) and residue characteristic \(p>0\). Let \((S,M,k)\) be a regular local ring of dimension three dominating \(A\), essentially of finite type over \(A\) with \(K:=QF(S)\) of transcendence degree two over \(F\). Let finally \(L/K\) be a finite field extension and \(v\) be a valuation of \(L\). Assume:
-
(i)
\(L/K\) is cyclic Galois or purely inseparable of degree \(p\).
-
(ii)
\(v\) has rank one and is centered in \(S\).
Then there exists a regular local ring \(T\) essentially of finite type over \(A\) with \(QF(T)=L\) such that \(v\) is centered in \(T\).
Applying embedded resolution techniques for surfaces, it can be assumed that such a \(v\) is centered in a local model of \(L\) of the form \(B=(S[X]/(h))_{(M,X)}\) with \(h\) monic of degree \(p\); more precisely, \(h=X^p-g^{p-1}X-f\), \(f,g \in M\) and (\(g=0\) if \(\text{ char}A=0\)). In particular the Local Uniformization statement of Conjecture 1.2 only involves certain hypersurface singularities \((\text{ Spec}\,B,x)\), of multiplicity \(m(x) \le p=\text{ char}k\), and embedded in an excellent fourfold \((Z=\text{ Spec}\,S[X],x)\). We prove here:
Main Theorem 1.3
Let \((R,\mathfrak M , k=k(x):={R \over \mathfrak{M }})\) be an excellent regular local ring of dimension four, \((Z,x):=(\text{ Spec}R,\mathfrak{M })\) and \((X,x):=(\text{ Spec}R/(h),x)\) be a reduced hypersurface. Assume that the multiplicity \(m(x)\) of \((X,x)\) satisfies \(m(x)< p:= \text{ char}k(x)\). Let \(v\) be a valuation of \(K(X)\) centered at \(x\). Then there exists a finite sequence of local blowing ups
where \(x_i\in X_i\), \(0\le i \le n\) is the center of \(v\), each blowing up center \(Y_i \subset X_i\) is permissible at \(x_i\) (in Hironaka’s sense), such that \(x_n\) is regular.
The proof of theorem 1.3 builds upon classical Resolution of Singularities techniques. We use systematically the Hironaka characteristic polyhedron and Hironaka’s invariants: the multiplicity \(m(x)\) and \(\tau \)-number \(\tau (x)\) for the hypersurface singularity \((X,x):=(\text{ Spec}R/(h),x)\).
Since we are working without any ground field (at least when \(R\) is not equicharacteristic), the Tschirnhausen trick (killing the degree (\(m(x)-1\))-term in the equation) cannot be directly applied even though \(m(x)<p\). Rather, we use it for the initial face of the Hironaka polyhedron (Theorem 3.5) to define well behaved invariants. The pair \((m(x),\tau (x)\)) is then further on completed to a 6-tuple \(\iota (x)\) defined in 5.2.
The main technical part is concentrated in Sect. 7. We consider projections to a two dimensional space to define a refined invariant in Sects. 7.3 and 7.4.1. Controlling the transformation law for this refined invariant under blowing up is much harder but leads essentially to the same formulæ as for the characteristic polygon of a surface singularity. The proof of the Main Theorem follows rather easily from these computations (Sect. 7.4.5).
It is worth pointing out that these techniques are global in nature and it is to be expected that Theorem 1.3 can be extended to a global version, i.e. without referring to a given valuation \(v\) and using global blowing up centers. We use the valuation only at a few specific places (mostly in Sect. 6) to make the argument quicker.
This article is organized as follows: Sect. 2 states the reduction of the Main Theorem to the case \(\tau (x)=1\), immediate from [11]. This means that the initial form \( \text{ in}_x(h)\) can be written
Section 3 first recollects known material from [16, 17] about characteristic polyhedra and associated invariants (Definition 3.2). Special coordinates \((z,u):=(z,u_1,u_2,u_3)\) on \(R\) are said to be fully prepared if they compute the Hironaka characteristic polyhedron \(\Delta (h;u)\) and if the \(\delta \)-initial \( \text{ in}_\delta (h)\) of \(h\) is Tschirnhausen transformed, i.e. has no term of degree \(m(x)-1\) (Theorem 3.5 and Definition 3.5.3). The form \( \text{ in}_\delta (h)\) is defined in 3.2(iii) and is the sum of the initial forms of all those terms in \(h\) contributing to the face of minimal order at \(x\) of \(\Delta (h;u)\). In such special coordinates, invariants \(d_1,d_2,d_3, \epsilon (x)\) (each of them is a nonnegative rational number) can be computed from the polyhedron \(\Delta (h;u)\) and the \(\delta \)-initial \(\text{ in}_\delta (h)\) (Definition 3.7). Theorem 3.9 proves the Main Theorem when \(\epsilon (x)=0\), in which case only combinatorial blowing ups are used.
When \(\epsilon (x)>0\), some preparations are required in order to get the locus
Zariski closed and of dimension at most one (Theorems 3.10 and 3.11). Section 4 prepares \( \text{ Spec}R\) and constructs an equicharacteristic \(p\) normal crossings divisor
which contains \(\Sigma \) (\(E=E_n\) in Proposition 4.1).
Section 5 then provides some further invariants build up from the ideal of coefficients of \( \text{ in}_\delta (h)\) once this preparation is achieved: a refined directrix \(\mathcal{V} \subseteq \langle U_1,U_2,U_3\rangle \) (Definition 5.1) and associated refined numerical invariant \(\iota (x)\) (Definition 5.2):
Section 6 introduces the notion of \(\epsilon \)-permissible blowing up centers (Definition 6.1). For curves, being \(\epsilon \)-permissible is stronger than being Hironaka permissible (Proposition 6.2); blowing up along an \(\epsilon \)-permissible center does not increase the invariant \(\iota (x)\) (Proposition 6.3). Furthermore, \(\iota (x)\) can be decreased by blowing up along \(\epsilon \)-permissible centers except possibly when \(\mathcal{V}=\langle U_3\rangle \) and (either \( \text{ div}(u_3)\subseteq E\) or \(E \subseteq \text{ div}(u_1u_2)\)) (Propositions 6.4 and 6.5).
Section 7 proves the same result in these remaining cases (Theorem 7.1), thus concluding the proof of the Main Theorem. We now project to the \((u_1,u_2)\)-space and define well prepared coordinates by minimizing the induced image of \(\Delta (h;u)\) by this projection (this requires choosing special coordinates \((z,u_3)\)). There are further associated invariants \(\beta (u,z), C(u,z),\gamma (u,z)\) defined in 7.4.1. The behaviour of these invariants by blowing up \(\epsilon \)-permissible curves and closed points are studied respectively in Propositions 7.4.2 and 7.4.3. Section 7.4.5 contains the proof of theorem 7.1 and is basically a consequence of the former computations.
The notation and assumption in the Main Theorem will be kept all along this article. The proof will be made by induction on the multiplicity \(m(x)=\text{ ord}_x(h)\) of \(x\in X\). Since it is assumed that \(m(x)<p, (X,x)\) is already regular if \(p=2\), so we assume \(p \ge 3\) from now on. The formal completion of \(R\) with respect to \(\mathfrak{M }\) is denoted by \(\widehat{R}\).
2 Basic invariants
Two basic invariants are attached to the hypersurface singularity \((X,x)=( \text{ Spec}R/(h),x)\). The first invariant is its multiplicity \(m(x)\) (or \(m\) for short) of \((X,x)\). The second invariant is \(\tau (x)\) (or \(\tau \) for short), which is the dimension of the smallest \(k(x)\)-vector subspace \(\mathcal{T}\) of \( {\mathfrak{M }\over \mathfrak{M }^2}\) such that in\(_x(h)\in k(x)[\mathcal{T}]\) [16], Ch.2, Lemma 10. This vector space is called the directrix of in\(_x(h)\).
Proving the Main Theorem in the cases dim\((Z)-\tau (x)\in \{0,1,2 \} \), i.e. \(\tau (x)\in \{2,3,4\} \) is done in [11]. So from now on, we assume that \(\tau (x)=1\). Equivalently:
3 Characteristic polyhedron
Definition 3.1
-
(i)
An \(F\)-subset \(\Delta \subset \mathbb R _{+}^{d}\) is a closed complex subset of \(\mathbb R _{+}^{d}\) such that \(v\in \Delta \) implies \(v+\mathbb R _{+}^{d} \subset \Delta \).
-
(ii)
A point \(v\in \Delta \) is called a vertex if there is a positive linear form \(L\) on \(\mathbb R ^{d}\) (i.e. has strictly positive coefficients) such that
$$\begin{aligned} \{v\}=\Delta \cap \{A\in \mathbb R ^d|L(A)=1\}. \end{aligned}$$ -
(iii)
The essential boundary \(\partial \Delta \) of an \(F\)-subset \(\Delta \) is the subset of \(\Delta \) consisting of those \(v\in \Delta \) such that \(v\not \in v^{\prime }+\mathbb R _{+}^{d} \) with \(v^{\prime }\in \Delta \) unless \(v^{\prime }=v\). We write \(\Delta ^+=\Delta - \partial \Delta \).
For the next definition and proposition, we will forget the hypothesis dim\((R)=4\): we will have to use the notions defined there for different regular rings of dimension at most three. Given a r.s.p. \((y,u_1,u_2,\dots ,u_d)=:(y,u)\) of a regular local ring \(R\) and \(f \in R\), there exists a finite sum expansion
where each \(C_{A,b}\) is a unit in \(R\). This follows easily from the facts that \(R\) is Noetherian and the map \(R \subseteq \widehat{R}\) faithfully flat. We regard \(u\) as “fixed” parameters and \(y\) as “varying”, which is reflected in the indexing below. Assume furthermore that
We let \(\overline{R}:=R/(u_1, \ldots ,u_d)\), \(\overline{h}\in \overline{R}\) be the image of \(h\) and “\(\text{ ord}\)” be the valuation of the discrete valuation ring \(\overline{R}\). We extend our conventions by letting now
Assumption (27*) and notation (3) are maintained all along this article. Our original concern is for \(\tau (x)=1\), say \(\text{ in}_x(h)=\lambda Y^m\), \(0 \ne \lambda \in k(x)\) which fits into these conventions provided \(Y=\text{ in}_x(y)\).
Definition 3.2
-
(i)
The polyhedron \(\Delta (h;u;y) \subset \mathbb R _{\ge 0}^{d}\) is defined as the smallest \(F\)-subset containing all points of
$$\begin{aligned} S(h):=\left\{ v={A \over m-b}| 0 \le b<m \right\} \!. \end{aligned}$$The characteristic polyhedron \(\Delta (h;u) \subset \mathbb R _{\ge 0}^{d}\) is defined by the formula
$$\begin{aligned} \Delta (h;u):=\bigcap _{(\widehat{y},u_1,\dots ,u_d)}\Delta (h;u;\widehat{y}), \end{aligned}$$(4)where the intersection runs over all r.s.p’s of \(\widehat{R}\) of the form \((\widehat{y},u_1,\dots ,u_d)\).
-
(ii)
For \(v\in \partial \Delta (h;u;y)\), the \(v\)-initial of \(h\) is defined as
$$\begin{aligned} \text{ in}_v(h):=\sum \limits _{A,b} \overline{C_{A,b}}Y^bU^A \in k[U,Y]=k[U_1,U_2,\dots ,U_d,Y], \end{aligned}$$where \( \overline{C_{A,b}}\in k\) is the residue of \( {C_{A,b}}\) and the sum ranges over such \((A,b)\) that
$$\begin{aligned} C_{A,b}\not =0, \ (b\le m, A=0) \ \text{ or}\ \left(b<m\ \text{ and}\ v={A \over m-b}\right). \end{aligned}$$ -
(iii)
For \(A\in \mathbb N ^{d}, let |A|:=a_1 + \cdots + a_{d}\). We put
$$\begin{aligned} \delta (h,u,y):=\text{ min}\left\{ {|A| \over m-b}:\ C_{A,b}\not =0,b<m \right\} \!. \end{aligned}$$This is in fact an invariant of the polyhedron \(\Delta (h;u;y)\) since
$$\begin{aligned} \delta (h,u,y)=\text{ min}\{|v|\ : \ v\in \Delta (h;u;y)\}. \end{aligned}$$The \(\delta \)-initial of \(h\) is defined as
$$\begin{aligned} \text{ in}_{\delta ,u,y}(h):=\sum \limits _{A,b} \overline{C_{A,b}}Y^bU^A \in k(x)[U,Y]=k(x)[U_1,U_2,\dots ,U_d,Y], \end{aligned}$$where the sum ranges over such \((A,b)\) that
$$\begin{aligned} C_{A,b}\not =0, \ (b\le m, A=0) \, \text{ or}\ \left(b<m\ \text{ and}\ {|A| \over m-b}=\delta (h,u,y)\right). \end{aligned}$$ -
(iv)
More generally, let
$$\begin{aligned}&L: \, (x_{1},x_{2},\dots ,x_{d})\mapsto L(x_{1},x_{2},\dots ,x_{d})=\lambda _{1}x_{1}+\lambda _{2}x_{2}+\dots +\lambda _{d}x_{d}, \, \lambda _{1},\lambda _{2},\dots ,&\qquad \lambda _{d}\in \mathbb Q _{\ge 0}, \end{aligned}$$be a nonzero nonnegative linear form on \(\mathbb R ^{d}\). We define
$$\begin{aligned} l (h,u,y):=\text{ min}\{{L(A) }|A\in \Delta (h;u;y)\}\ge 0. \end{aligned}$$We define a monomial valuation \(v_{L,h,u,y}\) on \(R\) by setting
$$\begin{aligned} I_\lambda := (\{y^bu^A| l(h,u,y)b+ L(A)\ge \lambda \})\subseteq R, \end{aligned}$$for \(\lambda \ge 0\) and \(v_{L,h,u,y}(g):=\text{ min}\{\lambda \in \mathbb Q | g \in I_\lambda \}\) for any nonzero \(g \in R\).
Proposition 3.3
Let \(L\) be a nonzero nonnegative linear form as above, and let
The graded algebra \( \text{ gr}_{v_{L,h,u,y}}(R)\) of \(R\) w.r.t. \(v_{L,h,u,y}\) is given by
-
(i)
if \(l(h,u,y)\not =0\), then
$$\begin{aligned} \text{ gr}_{v_{L,h,u,y}}(R)={R \over (y,\{u_i\}_{i\in I})}[Y,\{U_i\}_{i\in I}]; \end{aligned}$$ -
(ii)
if \(l(h,u,y)=0\), then
$$\begin{aligned} \text{ gr}_{v_{L,h,u,y}}(R)={R \over (\{u_i\}_{i\in I})}[\{U_i\}_{i\in I}]. \end{aligned}$$In particular, we have \(\text{ gr}_{v_{L,h,u,y}}(R)\simeq k[Y,U_1,U_2,\dots ,U_d]\) whenever \(L\) is positive.
The above proposition is obvious. One also checks easily the following:
Remark 3.3.1
Let \(v\) be a vertex of \(\Delta (h,u,y)\). We have:
-
(i)
\(\text{ in}_v(h)\) is independent of the presentation 3.1 (1),
-
(ii)
\(\text{ in}_v(h)\not = \text{ in}_x(h)\),
-
(iii)
\(L\) being a positive linear form \(L\) on \(\mathbb R ^d\) such that \(\{v\}=\Delta \cap \{A\in \mathbb R ^d|L(A)=1\}\), (cf. 3.1(ii)), then
$$\begin{aligned} \text{ in}_{v}(h)=\text{ in}_{{v_{L,h,u,y}}}(h)\in \text{ gr}_{v_{L,h,u,y}}(R)=k(x)[U_1,U_2,\dots ,U_d,Y]. \end{aligned}$$
When there is no ambiguity, we will write \( \text{ gr}_{\delta }(R)\) and \( \text{ in}_{\delta }(h)\in \text{ gr}_{\delta }(R)\) instead of respectively \(\text{ gr}_{v_{L,h,u,y}}(R)\) and \(\text{ in}_{{v_{L,h,u,y}}}(h)\), where \(L(x_1,x_2,\dots ,x_d)=x_1+x_2+\dots +x_d\).
Remark 3.4
With notations as above, we have: \(\delta (h,u,y)\in {1 \over m!}\mathbb N \) and \(\delta (h,u,y)>1\) if (\(m=\text{ ord}_x(h)\) and \(\langle \text{ in}_x(h)\rangle = \langle (\text{ in}_x(y))^m\rangle \)).
Assumption 3.4.1
We now apply these constructions to the case \(R:=\mathcal{O}_{Z,x}\), dim\((R)=4\); the element \(h\in R\) verifies assumptions 3.1(2)(3) with \(m=\text{ ord}_x(h)<p=\text{ char}k(x)\) and \(\langle \text{ in}_x(h)\rangle = \langle Y^m\rangle \). In addition, \(X=\text{ Spec}(R/(h))\) is reduced.
Theorem 3.5
Given \((y,u_1,u_2,u_3)=:(y,u)\) as above, there exists \(z \in R\), \(z\equiv y\) mod\((u_1,u_2,u_3)\) such that
Proof
Suppose \(\Delta (h,u) = \varnothing \), then, in \(\widehat{R}\), we should have \(h=\gamma z^m\), \(\gamma \) invertible in \(\widehat{R}\) and \(z \in \widehat{R}\) a local parameter: \(h\) should be nonreduced in \(\widehat{R}\). By excellence, \(h\) should be nonreduced in \( R\), in contradiction with the hypothesis \(X\) reduced.
Since \(\Delta (h,u) \not = \varnothing , \Delta (h,u)\) may be defined by a finite number \(n\) of inequalities \(L_i(x_1,x_2,x_3)\ge 1\), \(1\le i \le n\), with \(L_i(x_1,x_2,x_3)=a_{i,1}x_1+a_{i,2}x_2+a_{i,3}x_3\), \(a_{i,1},a_{i,2},a_{i,3}\in \mathbb Q _{\ge 0}, L_i\ne 0\). In a few words:
We choose \(L_1(x_1,x_2,x_3)={1 \over \delta (h,u)}(x_1+x_2+x_3)\), with \(\delta (h,u):=\text{ min}\{|v|: v\in \Delta (h,u)\}\).
Suppose (5) does not hold for \((y,u)\). Then, with notations as in 3.2(iv), some \(L_i\), \(1\le i \le n\) satisfies
We skip the index \(i\) of \(L_i\) and of \(l_i(h,u,y)\) to simplify the notations. Following 3.2(iv), we define the initial form of \(h\) with respect to \(L,u,y \): in the case \(l(h,u,y)>0\)
with \(bl(h,u,y)+L(A)=ml(h,u,y), \ \overline{C_{A,b}}\in {R \over (u_i)_{a_i>0}+(y)}\). In the case \(l(h,u,y)=0\), we have
Claim 3.5.1
In (7) (resp. (8)), there exists \(A\) with \(\overline{C_{A,m-1}}\not =0\in {R \over (u_{i})_{a_i>0}}\) (resp. \(\overline{C_{A,m-1}y^{m-1}}\not =0\in {R \over (u_{i})_{a_{i}>0}}\)).
Indeed, in the face with equation \(L(x_{1},x_{2},x_{3})=l(h,u,y)\) of \(\Delta (h,u,y)\) there is at least a vertex \(v\) which is solvable [17, (3.8)]. Then \(in_v(h)\) is collinear to an \(m\)th-power: \(C_{v,m-1}\not \in \mathfrak M \) since \(m<p\) and the claim is proved. Take \(A=v\) and let
Note that, for any \(A\) with \( C_{A,m-1}\not =0\), \(A\in \Delta (h,u,y)\). So, for any \(i,\ 1\le i \le n\), and any \(A\) with \( C_{A,m-1}\not =0\), \(L(A)\ge l(h,u,y)\). So if in the expansion of 3.2(1) we set
we get a new expansion
and \(D_{A,b}\not =0\ \Rightarrow \ {L(A)}+bl_i (h,u,y)\ge m\). So
Suppose \(l(h,u,y_1)= l(h,u,y)\). Then \(v_{L,u,y_1}(y)=v_{L,u,y_1}(y_1)= l(h,u,y)\),
where (11) is the expansion of (12). In (11), the terms with \(b=m-1\) are all zero; in (10), \( D_{A,m-1}\not = 0\) implies \(L(A)+l(h,u,y_1)b>m l(h,u,y_1)\). By the claim, \(l(h,u,y_1)=1\): a contradiction, hence
Note that (9) is independent of the linear form \(L\), so
By induction on the \(l_i(h,u,y)\)’s, we get (5). If (5) holds but not (6) for \(h,u,y\), we make the change of variable (7) and get (6).
Proposition 3.5.2
With notations as above, assume furthermore that \((z,u)\) and \((z^{\prime },u^{\prime })\) satisfy (5) of the previous theorem. Then \(\delta (h,u,z)=\delta (h,u^{\prime },z^{\prime })\).
Proof This is obvious if \(u_i=u^{\prime }_i\), \(1 \le i \le 3\). On the other hand, the condition
is preserved if \(z=z^{\prime }\) since \(\delta (h,u,z)>1\) (Remark 3.4). In particular we have
and we conclude by symmetry that \(\delta (h,u,z)= \delta (h,u^{\prime },z^{\prime })\).
Definition 3.5.3
When \(z \in R\) is such that (5) (resp. (6) holds), we say that \((z,u)\) is prepared (resp. \(\delta \)-prepared). If both of (5) and (6) hold for \(z\in R\), we say that \((z,u)\) is fully prepared. If there is no ambiguity on \(u=(u_1,u_2,u_3)\), we simply say that \(z\) is prepared, \(\delta \)-prepared, or fully-prepared.
If \((z,u)\) is prepared, the invariant \(\delta (h,u,z)\) will be henceforth denoted by \(\delta (x)\).
Theorem 3.6
Let \((y,u_1,u_2,u_3)=:(y,u)\) be as before and \(E\) be a normal crossings divisor \(E\subseteq \)div\((u_1u_2u_3)\subset Spec(R)\). For a component div\((u_i)\) of \(E\), we define
-
(i)
Let \(z\in R\) be such that \((z,u)=(z,u_1,u_2,u_3)\) is prepared. Then
$$\begin{aligned} d_i(u)\ge 1 \Leftrightarrow V(z,u_i)\ \text{ is} \text{ a} \text{ permissible} \text{ blowing} \text{ up} \text{ center} \text{ of} \ Spec(R / (h)). \end{aligned}$$(14) -
(ii)
We have \(d_i(u)>0\) if and only if \(Y_i:= V(h,u_i)_\mathrm{red}\) is a regular surface.
-
(iii)
Assume that \(d_i(u)>0\). Then \(d_i(u)=\delta (\eta _i)\), where \(\eta _i \in Z\) is the generic point of \(Y_i= V(h,u_i)_\mathrm{red}\) as above; in particular, \(d_i(u)\) is independent of the choice of a prepared \((z,u)=(z,u_1,u_2,u_3)\) containing \(u_i\).
Proof of (ii). Take \(z\in R\) such that \(\Delta (h,u)=\Delta (h,u,z)\), so \(h=z^m\)mod\((u_i)\) iff \(d_i(u)>0\) iff \(Y:= V(z,u_i)=V(h,u_i)_{red}\). We get
Proof of (i). In that hypersurface case, \(Y:= V(z,u_i)\in \text{ Spec}R\) permissible means \(h\in (z,u_i)^m\) which is equivalent to \(d_i(u)\ge 1\).
Proof of (iii). Let \(\eta \) be the generic point of \(Y= V(h,u_i)_{red}=V(z,u_i)\). The following equivalence is straightforward
If \(0<d_i (u) <1\), we obviously have \(d_i(u)=d_i(u,y)\) for every r.s.p. \((y,u_1,u_2,u_3)\) (with \(\langle \text{ in}_x(h)\rangle = \langle Y^m\rangle \)).
We turn to the case:
We claim that \(d_i(u)=\delta (\eta _i )\). As \(V(z,u_i)=V(h,u_i)_\mathrm{red}\), this will prove the invariance of \(d_i(u)\). We take \(i=1\) and write \(\eta \) for \(\eta _1\) in the following lemma.
Lemma 3.6.1
Let \((y,u_1,u_2,u_3)\) be as above and assume that \(d_1(u,y)>1\). Then
where the right hand side is computed w.r.t. the datum \((h) \subset R_{(y,u_1)}\). If furthermore \((y,u)\) is prepared, then
Proof By 3.1 (1), we have a finite expansion
that we rewrite as:
As \(d_1(u,y)>1\), we have \(b+a_1>m\) for all \(b<m\). Note that every \(\sum \nolimits _{a_2,a_3}C_{A,b}u_2^{a_2}u_3^{a_3}\) is invertible in \(R_{(y,u_1)}=\mathcal{O}_{Z,\eta }\). Then \(\delta (h,u_1,y)=\inf \{{a_1 \over m-b}\vert {b<m}\}=d_1(u,y)\): this is (15).
Now suppose that \(y,u_1,u_2,u_3\) is prepared. We claim that \(y,u_1\) is prepared w.r.t. the datum \((h) \subset R_{(y,u_1)}\). If not, then
is proportional to an \(m\mathrm{th}\)-power. So there exists some \(a_1\) with
so there exists some \(A\) with \(a_1+(m-1)d_1=d_1m\) and \(C_{A,m-1}\not =0\). Then, as in 3.5 (9), we change the variable \(y\) by \(y_1=y+{1\over m}C_{0,m}^{-1}\sum _{A}C_{A,m-1}u^A\in R\), we get a new expansion
Now \(D_{A,b}\not =0\) implies \({L_i(A)\over l_i (h,u,y)}+b\ge m\) for each linear form \(L_i\) such that
This holds in particular for the linear form \(L(x_1,x_2,x_3)={1\over d_1}x_1\). Since \(y\) is supposed to be nonprepared for \(u_1\), the unique vertex \(d_{1}(u_{1},y)\) of \(\Delta (h,u_{1},y)\subset \mathbb R ^{+}\) does not belong to \(\Delta (h,u_{1},y_{1})\). We get \(d_{1}(u_{1},y_{1})>d_{1}(u,y)\), a contradiction with the fact that \(\Delta (h,u,y)\) was minimal.
Definition and Notation 3.7
Given \((y,u_{1},u_{2},u_{3})=:(y,u)\), \(h\) reduced, with assumptions 3.1(2)(3) and a normal crossings divisor \(E\subset div(u_{1}u_{2}u_{3})\subset Spec(R)\), we let \(d_{i}:=d_{i}(u)\) for each irreducible component \( div(u_{i})\) of \(E\). We let \(d_i:=0\) whenever \(div(u_{i})\) is not an irreducible component of \(E\).
We define \(\epsilon (x,E)\in \mathbb Q _{\ge 0}\) (or \(\epsilon (x)\) for short) by:
These invariants appear in [13] Ch.1 (II.3.3) in an equal characteristic context. The following remarks are obvious from the definitions.
Remark 3.8
We have
-
(i)
\(\epsilon (x,E)\in {1\over m!}\mathbb N ,\)
-
(ii)
if \(\epsilon (x,E)=0\), \(\Delta (h,u)\) has only one vertex: the point \(v=(d_{1},d_{2},d_{3})\).
Theorem 3.9
Given \((y,u_{1},u_{2},u_{3})=:(y,u)\) and a divisor \(E\subset div (u_{1}u_{2}u_{3})\subset Spec(R)\) as above, assume that \(\epsilon (x,E)=0\). There exists a finite sequence of local blowing ups
where \(x_0=x\), \(x_{i}\in X_{i}\), \(0\le i \le n\) is the center of \(v\), each blowing up center \(Y_{i} \subset X_{i}\) is permissible in Hironaka’s sense, such that \(m(x_n)<m(x)\).
Proof See the connection with [13, Ch.1 (II.4.6)]. Let \(z \in R\) be such that \(\Delta (h,u,z)=\Delta (h,u)\). Then \(\delta (x)=\sum \nolimits _{1\le i \le 3}d_i\). Let \(\mathcal{I}\subset \{1,2,3\}\) satisfy the two following conditions:
P (for permissibility): \(\sum \nolimits _{i\in \mathcal{I}}d_i\ge 1\),
M (for maximality): \(|\mathcal{I}|\) minimal for P, i.e. the dimension of V\((z,\langle u_i,i\in \mathcal{I}\rangle )\) is maximal for P.
Note that \(\mathcal{I}\subset \{1,2,3\}\) is not unique in general. Then we choose \(\mathcal{I}\) with PM and we blow up \(Z\) along V\((z,\langle u_i,i\in \mathcal{I}\rangle )\). Let \(e: \ Z^{\prime } \longrightarrow Z\) denote this blowing up, \(X^{\prime }\) be the strict transform of \(X, x^{\prime }\in X^{\prime }\) be a point above \(x\), \(E^{\prime }\subset Z^{\prime }\) be the reduced inverse image of \(E\).
We claim that for either \((m(x^{\prime }),-\tau (x^{\prime }))<_{lex}(m(x),-\tau (x))\) or
Since \(\delta (h,u,y)\in {1 \over m!}\mathbb N \), a descending induction on \(\delta (x)\) reduces to \(m(x^{\prime })<m(x)\) or \((m(x^{\prime })=m(x), \tau (x^{\prime })\ge 2)\). As stated in Sect. 2, this completes the proof.
Proof of Claim We only treat the case \(\mathcal{I}= \{1,2,3\}\), the other cases being similar, if somewhat simpler. By PM, \(\mathcal{I}= \{1,2,3\}\) means
By [18, thm. 3, p. 331], if \(m(x^{\prime })=m(x)\), then \(x^{\prime }\) lies on the strict transform of \(z=0\). The variables \(u_1,u_2,u_3\) play symmetric roles; so after reordering, it can be assumed that \(x^{\prime }\) belongs to the affine chart \( \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\). Let
Let
be an expansion 3.1(1) of \(h\) with \((z,u)\) fully prepared (Theorem 3.5 and Definition 3.5.3),
where \(h^{\prime }_1 \in {I_{\delta (x)}^{\prime }}^+:= ({z^{\prime }}^{m+1}, {z^{\prime }}^b u_1^{(m-b)( \delta (x)-1)+1}, 0\le b \le m)\).
Since \(\epsilon (x)=0, C_{A,b}\) invertible in (18) implies \(a_2=(m-b)d_2, \ a_3=(m-b)d_3\). Note also that \(\delta (x)-1=d_1+d_2+d_3-1<d_1\) by (17). As \(d_1+d_2<1\) and \(d_1+d_3<1\), \(m(x^{\prime })=m(x)\) implies \(x^{\prime }=(z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3)\). The coordinate change \((z,u_1,u_2,u_3)\mapsto (z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3)\) is a monomial substitution, so \(\Delta (h^{\prime },u^{\prime },z^{\prime })\) is again minimal. With natural notations, we get \((d^{\prime }_1,d^{\prime }_2,d^{\prime }_3)=(\delta (x)-1,d_2,d_3),\epsilon (x^{\prime })=0\) and
provided \(\langle \text{ in}_{x^{\prime }}(h^{\prime })\rangle =\langle {Z^{\prime }}^2\rangle \), i.e. \(\tau (x^{\prime })=\tau (x)=1\).
Theorem 3.10
Given \((y,u_1,u_2,u_3)=:(y,u)\) and a divisor \(E\subset \,div(u_1u_2u_3)\subset Spec(R)\) as above, assume that \(E\) is equicharacteristic \(p= \text{ char}k(x)\) and
Then the set \(\{y\in X|m(y)=m(x),\epsilon (y)>0\ \text{ and}\ \tau (y)=1\}\) is locally closed.
Proof It is well known that the set
is locally closed. Suppose \(\epsilon (x)=0\) for some closed point \(x \in E\). We choose a r.s.p. \((z,u_1,u_2,u_3)\) of \(R\) at \(x\) which is fully prepared. There is a finite expansion
with each \(C_{A,b}\) invertible in \(R\) and \({a_i }\ge bd_i\), \(i=1,2,3\). Since \(\epsilon (x)=0\), there exists \(C_{A,b}\) such that \({a_i }= bd_i\), \(i=1,2,3\), with \(b\ge 2\) by full preparedness. Then the locus
contains the intersection of Sing\(_{m(x)}(X)\) with the complement of the hypersurface \(V(C_{A,b})\).
Theorem 3.11
With assumptions as in 3.10, assume furthermore \(\epsilon (x)>0\) and let \(F\) be an irreducible component of \(E\) with \(x \in F\). Then
Proof Say \( \text{ div}(u_1)\) is the given component. If \(d_1<1\), then \( \text{ dim}(\{y\in X|m(y)=m(x)\}\cap F)\le 1\) and the result is clear.
Assume now \(d_1>1\) and pick a fully prepared \((y,u_1)\) w.r.t. to datum \(h \subset R_{y,u_1}\). There exists a nonempty Zariski open set \(\Omega \subseteq F\) such that for \(y \in \Omega \) there is an expansion
with \(a_i \ge id_1\), \(\gamma _i \in \mathcal{O}_{X,y}\) for \(i \ge 1\). By definition of \(d_1\), some \(i\ge 2\) satisfies (\(a_i=id_1\) and \(\gamma _i \not \in (u_1)\)). By full preparedness, we have \(\gamma _1 \in (u_1)\) if \(a_1=d_1\). Let \(\Omega ^{\prime }\) be the intersection of \(\Omega \) with complement of the proper closed subset \(V(\gamma _i)\), so
Assume finally \(d_1=1\). The same construction now yields
4 Construction of the divisor E
In this section, we reach the assumptions of 3.10. We show they are stable under a class of local permissible blowing ups which we will prove ahead are sufficient to prove Theorem 1.3. We stick to Assumptions 3.4.1.
Proposition 4.1
With assumptions as above, there exists a finite sequence of local blowing ups
where \(x_0=x\), \(x_i\in X_i\) (\(X_i\) denoting the strict transform of \(X\)), \(0\le i \le n\), is the center of \(v\), each blowing up center \(Y_i \subset X_i\) is permissible for \(X_i\) in Hironaka’s sense, such that one the following properties holds:
-
(i)
\((m(x_n),-\tau (x_n))<_{lex}(m(x),-\tau (x))\), or
-
(ii)
\((m(x_n),-\tau (x_n))=(m(x),-\tau (x))\) and there exists a (reduced) normal crossings divisor \(E_n\subset (Z_n,x_n)\) of equicharacteristic \(p= \text{ char}k(x_n)= \text{ char}k(x)\) such that
$$\begin{aligned} C_{x_n}(E_n)\bot C_{x_n}(X_n)_{red}, \end{aligned}$$(20)$$\begin{aligned} S_n:=\text{ Sing}_{m(x_n)}\subset E_n, \end{aligned}$$(21)
where \(C_{x_n}\) denotes the tangent cone and \( \text{ Sing}_{m(x_n)}(X_n)\) is the stalk at \(x_n\) of the set of multiplicity \(m(x)\).
Proof We begin with the following lemma.
Lemma 4.2
With assumptions as above, assume furthermore that there exists a normal crossings divisor \(E\subset (Z,x)\) such that
Then for any local blowing up:
of center \(Y\subset X\), permissible for \((X,x)\) and at normal crossing with \(E\), we have \((m(x^{\prime }),\tau (x^{\prime }))\le (m(x),\tau (x))\), where \(x^{\prime }\in X^{\prime }\) is the center of \(v\); if equality holds, then
where \(E^{\prime }:=\pi ^{-1}(E)_{red}, X^{\prime }\) the strict transform of \(X\).
Proof The normal crossing assumption implies that we can choose a r.s.p. \((v_1,v_2,v_3,v_4)\) of \(R:=\mathcal{O}_{Z,x}\) such that \(Y=V(v_{1},\ldots ,v_{e})\) and \(E \subseteq \text{ div}(v_1 \cdots v_4)\). By permissibility, we have \(h\in (v_1,\ldots ,v_e)^{m(x)}\). Assumption (22) means that \(\langle \text{ in}_x(h)\rangle =\langle Z^{d}\rangle \), where \(Z \not \in \langle \text{ in}_x(v_j), \ \text{ div}(u_j) \subseteq E\rangle \). Changing generators of the ideal of \(Y\), we relabel parameters as \((z,u_1,u_2,u_3)\) with
If \(m(x^{\prime })=m(x), x^{\prime }\) belongs to the strict transform of \(\text{ div}(z)\) by (22). Let \(i\in A\) such that \(u_i\) generates the ideal of the exceptional divisor of \(\pi \) in a neighbourhood of \(x^{\prime }\) and let \(z^{\prime }=z/u_i\). A local equation for \((X^{\prime },x^{\prime })\) is \(h^{\prime }:=h/u_i^{m(x)}\), where
and
which proves the lemma.
Proof of 4.1 If \(\text{ Sing}_{m(x)}(X)=\{x\}\), take \(E=\text{ div}(u_1u_2u_3)\) with coordinates as in (23) above.
If \(\dim ( \text{ Sing}_{m(x)}(X))\ge 1\), then any regular closed set \(Y\subset S:=S_0\) is permissible for \(X\). In any case, we have \((m(x_n),-\tau (x_n)) \le (m(x),-\tau (x))\) since centers are permissible.
Let \((C,x)\) be any curve contained in \(S\). Since it is assumed that \(v\) has rank one, \(x_n\) does not belong to the strict transform \(C_n\) of \(C\) in \(X_n\) for \(n\gg 0\) if we take \(Y_i=\{x_i\}\), the center of \(v\) in \(X_i\) for \(i \ge 0\). In particular, it can be assumed that the strict transform \(S_n\) of \(S\) in \(X_n\) has pure dimension two. Take \(n=0\) in what follows.
We now apply classical embedded resolution theorems for \(S\) with \(\text{ dim}(S)= 2\) ([11] for suitable generality). This involves blowing up closed points or regular curves on the successive strict transforms of \(S\). By blowing up finitely closed many points as before, it can be assumed that every blown up curve is equicharacteristic \(p= \text{ char}k(x)\). We reach the following situation: the strict transform \(S_n\) of \(S\) at \(x_n\) is empty or an irreducible surface with normal crossings with the (equicharacteristic) reduced exceptional divisor \(E_n\) of \((Z_n,x_n)\rightarrow (Z,x)\). If \(S\) itself is equicharacteristic, enlarge \(E_n\) to \(E_n \cup S\). Otherwise, we blow up finitely many times irreducible components of \(S \cap E_n\) (i.e. equicharacteristic curves) to get \(x_n \not \in S_n\). This is possible again because \(v\) has rank one.
5 Refined directrix, transverseness, encombrement
Assume that the conclusion of Proposition 4.1 (ii) holds. We will perform local blowing ups which are permissible in Hironaka’s sense, with center \(Y_n\) having normal crossing with \(E_n\). Take \(n=0\) in what follows, \(E=E_0\), and consider a local blowing up:
of center \(Y\subset X\), permissible for \((X,x)\) and at normal crossing with \(E\). We assume that
where \(x^{\prime }\in X^{\prime }\) is the center of \(v\). By Lemma 4.2, we have
where \(E^{\prime }:=\pi ^{-1}(E)_{red}\), \(X^{\prime }\) the strict transform of \(X\) in \(Z^{\prime }\).
Definition and Notation 5.1
Let \((z,u)\) be fully prepared with \(E\subset \)div\((u_1u_2u_3)\subset \text{ Spec}(R)\) as above. Let
where \(Z= \text{ in}_{\delta }(z)\), \(U_i= \text{ in}_{\delta }(u_i)\), \(1\le i \le 3\) (notations of 3.3.1). Each \(F_j\) is zero or homogeneous of degree \(j\delta (x)\); we have \(F_j=0\) if \(j\delta (x)\not \in \mathbb N \).
We define the refined tangent ideal of \(X\) at \(x\) as the ideal
We define the refined directrix of \(X\) at \(x\) as the smallest vector subspace \(\mathcal{V}\subseteq \langle U_1,U_2,U_3 \rangle \) such that
Let \(\rho (x):=\text{ dim}(\mathcal{V})\).
Remark 5.1.1
The following holds:
-
(i)
\(U_i^{\lceil jd_i\rceil }\) divides \(F_j\), \( 2\le j \le m\), for \(1\le i \le 3\) such that \( \text{ div}(u_i)\subset E\).
-
(ii)
\(I_x\) and \(\mathcal V\) do not depend upon choices of \(z,u\) satisfying the assumptions.
-
(iii)
the polynomials
are zero or homogeneous of degree \(m!\epsilon (x)\).
Statement (i) is a consequence of the definition of \(d_i\). For (ii), suppose \((z^{\prime },u^{\prime })\) is fully prepared, where \(E\subset \div (u^{\prime }_1u^{\prime }_2u^{\prime }_3)\). Let
for some matrix \((a_{i,j})\in \text{ GL}(3,R)\). Since \(\tau (x)=1\), we have
in \(\text{ gr}_\delta \). Computing w.r.t. the r.s.p. \((z,u^{\prime })\), we get
with \(F^{\prime }_j(U^{\prime }_1,U^{\prime }_2,U^{\prime }_3)=F_j(\overline{\mathcal{M}}.(U^{\prime }_1,U^{\prime }_2,U^{\prime }_3))\), \(\overline{\mathcal{M}}\) being the residue of \(\mathcal{M}\) in \( \text{ GL}(3,k(x))\). Since \((z^{\prime },u^{\prime })\) is fully prepared, no term in \(Z^{m-1}\) occurs in \( \text{ in}_\delta (h) \in k(x)[Z^{\prime },U^{\prime }_1,U^{\prime }_2,U^{\prime }_3]\) and this implies that \(\langle \text{ in}_\delta (Z)\rangle =\langle \text{ in}_\delta (Z^{\prime })\rangle \).
Statement (iii) immediately follows from definition 3.7.
Definition 5.2
Let \(E\) be a fixed normal crossings divisor and \((z,u)\) be fully prepared (always with the condition \(E\subset \text{ div}(u_1u_2u_3)\subset \text{ Spec}(R)\)) as above.
We call “transverseness” index of \(x\), denoted by \(t(x)\), the maximal dimension of a subspace of \(\mathcal V\) which is transverse to \(\langle U_i,\ \text{ div}(u_i)\subset E\rangle \). This is independent of the choice of a fully prepared r.s.p. \((z,u)\) by Remark 5.1.1(ii).
We call “encombrement” of \(x\), denoted by \(e(x)\), the minimum number of \(U_i\)’s among all possible fully prepared \((z,u)\) necessary to write a basis of \(\mathcal V\).
We define an invariant
For convenience, we extend the definition when \(\tau (x)\ge 2\) by letting \(\iota (x):=(m(x),-\tau (x),0,0,0)\), Theorem 1.3 being already proved in this special case (Sect. 2). Note that \(\epsilon (x)\in {1 \over m(x)!}\mathbb N \), so any decreasing sequence of values (for the lexicographical ordering) taken by \(\iota \) is finite.
Examples 5.2.1
Assume \(p\ge 5\).
-
(i)
\(h=z^3+u_1^2(u_1+u_2+u_3)^2,\ E= \text{ div}(u_1)\). Then \(\delta (x)={4\over 3}, d_1(x)={2\over 3},\mathcal{V}=\langle U_1+U_2+U_3\rangle , \rho (x)=1, t(x)=1, e(x)=1\): take the r.s.p. \((z,u_1,u_1+u_2+u_3,u_3)\).
-
(ii)
\(h=z^3+u_1^2(u_1+u_2+u_3)^2,\ E= \text{ div}(u_1u_2u_3)\). Then \(\delta (x)={4\over 3}\), \(d_1(x)={2\over 3}\), \(d_2(x)=d_3(x)=0\), \(\mathcal{V}=\langle U_1+U_2+U_3\rangle \), \(\rho (x)=1\), \(t(x)=0\), \(e(x)=3\): the only choice allowed upon \((u_1,u_2,u_3)\) is permuting or multiplying by a unit in \(R\).
-
(iii)
\(h=z^3+u_1^2(u_1^2+u_2^2),\ E= \text{ div}(u_1)\). Then \(\delta (x)={4\over 3}\), \(d_1(x)={2\over 3}, \mathcal{V}=\langle U_1,U_2\rangle \), \(\rho (x)=2\), \(t(x)=1, e(x)=2\).
-
(iv)
\(h=z^3+u_1^2(u_1^2+u_2^2),\ E= \text{ div}(u_1u_2)\). Then \(\delta (x)={4\over 3}, d_1(x)={2\over 3}, d_2(x)=0, \mathcal{V}=\langle U_1,U_2\rangle , \rho (x)=2, t(x)=0, e(x)=2\).
Remark 5.2.2
The French “encombrement” was proposed by J. Giraud twenty years ago (English: “cumbersomeness index” roughly).
6 Permissible blowing ups, behaviour of the invariants
We stick to the assumptions of the previous section and assume furthermore that \(\epsilon (x)>0\).
Definition 6.1
An \(\epsilon \)-permissible center (permissible center for short) \(Y\) at \(x\) is one of the following:
-
(i)
either \(Y:=\{x\}=V(z,u_1,u_2,u_3)\),
-
(ii)
or \(Y:=V(z,u_1,u_2)\) with \((z,u)\) fully prepared,
and
where \(L\) denotes the linear form \(L(x_1,x_2,x_3)={x_1+x_2 \over d_1+d_2+\epsilon (x)}\) (Definition 3.2).
Proposition 6.2
An \(\epsilon \)-permissible center at \(x\) is permissible in Hironaka’s sense.
Proof Indeed, we have just to look at the case of a curve V\((z,u_1,u_2)\). In that latter case, as \(d_1+d_2+\epsilon (x)\ge 1\), we have
where \(\eta \) is the generic point of V\((z,u_1,u_2)\), so
which means exactly that V\((z,u_1,u_2)\) is permissible in Hironaka’s sense.
‘
Proposition 6.3
Let \(\pi :Z^{\prime } \longrightarrow Z \) be the blowing up along an \(\epsilon \)-permissible center \(Y\) at \(x\), \(X^{\prime }\) be the strict transform of \(X\) (with transformed equation \(h^{\prime }\) at the center \(x^{\prime }\in X^{\prime }\) of \(v\)). We have:
-
(i)
\(\iota (x^{\prime }) \le \iota (x)\) (Definition 5.2). If equality holds (in which case we say that \(x^{\prime }\) is “very near” \(x\)), then \(E^{\prime }:=\pi ^{-1}(E)_{red}\) is transverse to the directrix \(\mathcal{T}^{\prime }\) of \(X^{\prime }\) at \(x^{\prime }\).
-
(ii)
if \(Y=\{x\}\) and \((m(x^{\prime }),-\tau (x^{\prime }),\epsilon (x^{\prime })) = (m(x),-\tau (x),\epsilon (x))\), then \(x^{\prime }\) lies on
with notations as in 5.1. The refined directrix \(\mathcal{V}^{\prime }\) at \(x^{\prime }\) satisfies \(\mathcal{V}^{\prime } \equiv U^{-1}\mathcal{V} \ \text{ mod}\langle U\rangle \) where \(U= \text{ in}_{x^{\prime }}(u)\), \(u\) an equation of the exceptional divisor of \(\pi \).
Proof First assume that \(Y=\{x\}\). By [18, thm.3, p. 331], if \(m(x^{\prime })=m(x)\), then \(x^{\prime }\) lies on the strict transform of \(z=0\). The variables \(u_1,u_2,u_3\) play symmetric roles; o after reordering, it can be assumed that \(x^{\prime }\) belongs to the affine chart \( \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\). Let
We can choose a r.s.p. at \(x^{\prime }\) in the following way: if \(x^{\prime }\) is the origin, take \((z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3)\); f \(x^{\prime }\) belongs to the strict transform of, say \( \text{ div}(u_2)\), we can take \((z^{\prime },u^{\prime }_1,u^{\prime }_2,v_3)\) with \(v_3=\sum _a \lambda _a {u^{\prime }_3}^a\), \(\lambda _a\in R\) a unit or zero (the sum is finite) whose residue \(\sum _a \overline{\lambda _a} {U_3}^a\in k(x)[U_3]\) is an irreducible polynomial; in the general case \(u^{\prime }_2(x^{\prime })u^{\prime }_3(x^{\prime })\not =0\), we take \((z^{\prime },u^{\prime }_1,v_2,v_3)\) where \(v_c=\sum _{a,b} \lambda _{a,b,c} {u^{\prime }_2}^a{u^{\prime }_3}^b\), \(c=2,3\) (sums are finite), \(\lambda _{a,b,c}\in R\) a unit or zero, and
is a maximal ideal. Let
with \(a(i,j)\ge jd_i, {2\le j \le m}\), \({1\le i \le 3}\), \(G_j \in k(x)[U_1,U_2,U_3]\) homogeneous, \(G_j=0\) or \( \text{ deg}(G_j)=j\delta (x)-(a(1,j)+a(2,j)+a(3,j))\) and \(G_j\) not divisible by \(U_i\), \(1\le i \le 3\).
Let \(h^{\prime }:=h/u_1^m\) define the strict transform of \(h\). We define the linear form
with associated valuation \(v^{\prime }:=v_{L^{\prime },h^{\prime },u^{\prime },z^{\prime }}\) (Definition 3.2). We have
where \( \text{ gr}_{v^{\prime }}(R^{\prime })=R^{\prime }/(z^{\prime },u^{\prime }_1)[Z^{\prime },U^{\prime }_1]\). Here, the meaning of \( G_j(1,u^{\prime }_2,u^{\prime }_3)\) is given by the inclusion
By (27), \(x^{\prime }_1=\delta (x) - 1\) is the minimum value of the first coordinate of points in \(\Delta (h^{\prime },u^{\prime }_1,v_2,v_3,z^{\prime })\). Since \(z\) is \(\delta \)-prepared, no vertex of \(\Delta (h^{\prime },u^{\prime }_1,v_2,v_3,z^{\prime })\) with first coordinate equal to \(x^{\prime }_1=\delta (x) - 1\) is solvable. We get
and for at least one vertex \((x^{\prime }_1=\delta (x)-1, x^{\prime }_2,x^{\prime }_3)\), we have
In case \(x^{\prime }\) belongs to the strict transform of some \( \text{ div}(u_i)\), \(i=2,3\), we have \(d_i(x^{\prime })=d_i(x)\) for \(u^{\prime }_i(x^{\prime })=0\) by 3.6(iii). This leads to:
with the convention: \( \text{ ord}_{x^{\prime }}({u^{\prime }_i}^{a})=0\) when \(a\in \mathbb Q _+\) and \(u^{\prime }_i(x^{\prime })\not =0\), \( \text{ ord}_{x^{\prime }}({u^{\prime }_i}^{a})=a\) when \(u^{\prime }_i(x^{\prime })=0\). This proves \((-\tau (x^{\prime }),\epsilon (x^{\prime })) \le (-\tau (x),\epsilon (x))\).
Assume that \((-\tau (x^{\prime }),\epsilon (x^{\prime })) = (-\tau (x),\epsilon (x))\). Then
for each \(j\) with \(F_j\not =0\). By [18, Theorems 2 and 3], this means that \(x^{\prime }\) lies on
This proves the first assertion of (ii) in this case. All other assertions are easy consequences of (28) and of its explicitation (29).
Assume now that \(u^{\prime }_2(x^{\prime })u^{\prime }_3(x^{\prime })\not =0\). If \(x^{\prime }\) is rational over \(x\), i.e. \(u^{\prime }_2(x^{\prime })=\lambda \in k(x),u^{\prime }_3(x^{\prime })=\mu \in k(x)\), we have
where \(I_{x^{\prime }}\) is the refined tangent ideal of \(x^{\prime }\) (cf. 5.1(ii)). This proves the last assertion of (ii) in this case. Finally, if \(x^{\prime }\) is not rational over \(x\), then dim\((\mathcal{V})=1\). We get
If \(b\not =0\), we take \(v_2:=a+bu^{\prime }_2+cu^{\prime }_3\)mod\((u_1)\) and we get by 5.1.1
which proves the last assertion of (ii) in this case. All other assertions are easy as in the previous case.
We now consider blowing up along a curve \(Y=V(z,u_1,u_2)\).
By [18, thm.3, p. 331], if \(m(x^{\prime })=m(x)\), then \(x^{\prime }\) lies on the strict transform of \(z=0\). The variables \(u_1,u_2\) play symmetric roles; so after reordering, it can be assumed that \(x^{\prime }\) belongs to the affine chart \(\text{ Spec}R[z/u_1,u_2/u_1,u_3]\subset Z^{\prime }\). Let
We can choose a r.s.p. at \(x^{\prime }\) in the following way: if \(x^{\prime }\) is the origin, take \((z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3)\); otherwise take \((z^{\prime },u^{\prime }_1,v_2,u^{\prime }_3)\) where \(v_v=\sum _a \lambda _a {u^{\prime }_3}^a\), \(\lambda _a\in R\) a unit or zero (the sum is finite) whose residue \(\sum _a \overline{\lambda _a} {U_3}^a\in k(x)[U_3]\) is an irreducible polynomial.
With notations (26), since V\((z,u_1,u_2)\) is \(\epsilon \)-permissible, we have
Let \(h^{\prime }:=h/u_1^m\) define the strict transform of \(h\). Equation (27) gets replaced by
with \(\Gamma _j \in R^{\prime }/(z^{\prime },u^{\prime }_1)\) whose residue in \(R^{\prime }/(z^{\prime },u^{\prime }_1,u^{\prime }_3)\) is \(G_j(1,u^{\prime }_2)\) for \(2 \le j \le m, \Gamma _0\) a unit.
By (27*), \(\delta (x)-1- d_3\) is the minimum value of the first coordinate of points in \(\Delta (h^{\prime },u^{\prime }_1,v_2,u^{\prime }_3,z^{\prime })\). As in (26) there is no \({Z^{\prime }}^{m-1}\), each vertex of \(\Delta (h^{\prime },u^{\prime }_1,v_2,u^{\prime }_3,z^{\prime })\) with first coordinate equal to \(\delta (x) - 1 -d_3\) is not solvable. Then
and for at least one vertex \((x^{\prime }_1=\delta (x)-1 -d_3, x^{\prime }_2,x^{\prime }_3)\), we have
By Theorem 3.6(iii), we have \(d_3(x^{\prime })=d_3\). If \(u^{\prime }_2(x^{\prime })\not =0\), this gives
If \(u^{\prime }_2(x^{\prime })=0\), we also have \(d_2(x^{\prime })=d_2\) by 3.6(iii). This leads to:
with the convention: \(\text{ ord}_{x^{\prime }}({u^{\prime }_i}^{a})=0\) when \(a\in \mathbb Q ^+\) and \(u^{\prime }_i(x^{\prime })\not =0, \text{ ord}_{x^{\prime }}({u^{\prime }_i}^{a})=a\) when \(u^{\prime }_i(x^{\prime })=0\).
If \(\epsilon (x^{\prime })=\epsilon (x)\), then
for each \(j\) with \(G_j\not =0\). By [18, Theorems 2 and 3], this means that \(x^{\prime }\) lies on
where the latter is identified with \(\pi ^{-1}(x)\subset Z^{\prime }\). The proof now runs parallel to the case \(Y=\{x\}\).
Proposition 6.4
With assumption as in 6.3, assume \(e(x)=3\) and \(Y=\{x\}\). Then \(x^{\prime }\) is not very near \(x\), i.e. \(\iota (x^{\prime })< \iota (x)\) (Definition 5.2).
Proof
If \(\rho (x)=3\), this follows from 6.3(ii), since \( \text{ Proj}k(x)[Z,U_1,U_2,U_3]/(Z,\mathcal{V})=\varnothing \).
When \(\rho (x)=2\), we have \(t(x)<2\) necessarily: otherwise we should have
By a linear change on the free variables \((u_2,u_3)\), we would get
i.e. \(e(x)=2\), a contradiction.
When \(\rho (x)=2\), \(t(x)=1\), we can choose parameters such that
By Proposition 6.3(ii), we have
with \(E^{\prime }= \text{ div}(u^{\prime }_1)\), \(V^{\prime }_2= \text{ in}_{x^{\prime }}(a+u_2/u_1)\), where \(a\in R\) is a preimage of \(\alpha \). Then \(\rho (x^{\prime })\ge t(x^{\prime })\ge 2\).
When \(\rho (x)=2\), \(t(x)=0\), then, up to a permutation on \(u_1,u_2,u_3\), we have \(E= \text{ div}(u_1u_2u_3)\) and
By Proposition 6.3(ii), we can take \(\pi ^{-1}(x)= \text{ div}(u_2)\) locally at \(x^{\prime }\), and r.s.p.
where \(a,b\in R\) are preimages of \(\alpha ,\beta \). In particular we get \(E \subseteq \text{ div}(u^{\prime }_2u^{\prime }_3)\). On the other hand, we have
and this proves that \(t(x^{\prime })\ge 1\).
When \(\rho (x)=1\), then \(e(x)=3\) implies \(E= \text{ div}(u_1u_2u_3)\) (so \(t(x)=0\)) and
up to renumbering parameters. By Proposition 6.3(ii), we can choose \(E^{\prime }\subseteq \text{ div}(u^{\prime }_1u^{\prime }_2)\), say \(\pi ^{-1}(x)= \text{ div}(u_1)\) locally at \(x^{\prime }\) and r.s.p.
at \(x^{\prime }\), where \(a,b\in R\) are preimages of \(\alpha ,\beta \). Since \( V^{\prime }_3 \in \mathcal{V^{\prime }} \oplus \langle U_1\rangle \), we get \(t(x^{\prime })\ge 1\).
Proposition 6.5
Let \(x\) satisfy the conclusion of Proposition 4.1(ii) and assume \(e(x)=2\). There exists a finite sequence of local blowing ups
where \(x_0=x\), \(x_i\in X_i\) (\(X_i\) denoting the strict transform of \(X\)), \(0\le i \le n\), is the center of \(v\), each blowing up center is \(Y_i =\{x_i\}\) such that \(\iota (x^{\prime })<\iota (x)\).
Proof
First assume that \(\rho (x)=1\). Then \(t(x)=0\), \( \text{ div}(u_2u_3)\subseteq E\) and we have \(\mathcal{V}=\langle \alpha U_2+U_3\rangle , \alpha \in k(x)^{\times }\) after possibly renumbering parameters. If \(x^{\prime }\) does not belong to the strict transform of \( \text{ div}(u_2)\), we can take \(\pi ^{-1}(x)= \text{ div}(u_2)\) locally at \(x^{\prime }\), and r.s.p.
where \(a\in R\) is a preimage of \(\alpha \). In particular we get \(E^{\prime } \subseteq \text{ div}(u^{\prime }_1u^{\prime }_2)\), with \(u^{\prime }_1=u_1/u_2\). On the other hand, we have
whence \(t(x^{\prime })\ge 1\), so \(\iota (x^{\prime })<\iota (x)\). Assume now that \(x^{\prime }\) belongs to the strict transform of \( \text{ div}(u_2)\). We can take \(\pi ^{-1}(x)= \text{ div}(u_1)\) locally at \(x^{\prime }\), and r.s.p.
We get \(E = \text{ div}(u^{\prime }_1u^{\prime }_2u^{\prime }_3)\) and
If \(\iota (x^{\prime })=\iota (x)\), then \(\mathcal{V^{\prime }}= \langle \alpha U^{\prime }_2+U^{\prime }_3\rangle \) and iterate. Since the valuation \(v\) has rank one, say \(v(u_2)<n v(u_1)\) for some \(n>0\), the process stops after iterating \(n\) times.
Assume that \(\rho (x)=2\). Then \(\mathcal{V}=\langle U_2,U_3\rangle \) after possibly renumbering parameters. We can take \(\pi ^{-1}(x)= \text{ div}(u_1)\) locally at \(x^{\prime }\) and r.s.p.
We get \( \text{ div}(u^{\prime }_1) \subseteq E^{\prime }\) and
hence \(\langle U^{\prime }_2+\alpha _2U^{\prime }_1 ,U^{\prime }_3+\alpha _3 U^{\prime }_1\rangle \subseteq \mathcal{V^{\prime }}\) for some \(\alpha _2,\alpha _3 \in k(x)\). If \(\iota (x^{\prime })=\iota (x)\), then equality holds; moreover \(\alpha _i=0\) whenever \( \text{ div}(u_i)\subseteq E\), \(i=2\) or \(i=3\).
Iterating, there exists a regular formal curve \(\widehat{Y} \subset X\) passing through all points \(x, x_1:=x^{\prime }, \ldots , x_n\), taking \(Y_i=\{x_i\}\) for each \(i\ge 0\). By standard arguments, \(\widehat{Y}\subseteq \text{ Sing}_{m(x)}(X)\). Our assumptions (beginning of Sect. 5) force \(\widehat{Y} \subset E\), say \(\widehat{Y} \subset \text{ div}(u_2)\). One concludes as in the case \(\rho (x)=1\).
7 Proof of the main theorem
By Theorem 3.9, a reduction in \(m=m(x)\) can be achieved when \(\epsilon (x,E)=0\) for some normal crossings divisor \(E \subseteq \text{ div}(u_1u_2u_3)\). The previous section (Propositions 6.4 and 6.5) reduces Theorem 1.3 to the only case \(\epsilon (x)>0\), \(e(x)=1\). There remains to prove the following:
Theorem 7.1
Let \(x\) satisfy the conclusion of Proposition 4.1(ii) (w.r.t. \(E \subseteq \text{ div}(u_1u_2u_3))\) and assume \(\epsilon (x)>0\), \(e(x)=1\). There exists a finite sequence of local blowing ups
where \(x_0=x\), \(x_i\in X_i\) (\(X_i\) denoting the strict transform of \(X\)), \(0\le i \le n\), is the center of \(v\), each blowing up center \(Y_i \subset X_i\) is permissible in Hironaka’s sense, such that
-
(i)
\(\iota (x_n)<\iota (x)\), and
-
(ii)
\(x_n\) satisfies the conclusion of Proposition 4.1(ii) (w.r.t. the strict transform \(E_n\) of \(E\) in \(Z_n)\) if \(((m(x_n),\tau (x_n))=(m(x),\tau (x))\) and \(\epsilon (x_n)>0\)).
The proof is long, needing new invariants and the control of their behavior under permissible blowing ups. There are two different cases:
-
(i)
\( t(x)=0,\ e(x)= 1\),
-
(ii)
\( t(x)=e(x)=1\).
In both cases, we choose the indices so that \(\mathcal{V}=\langle U_3\rangle \). We assume that
(P1) \((z,u)\) is fully prepared, and
(P2) \(E\subseteq \text{ div}(u_1u_2u_3)\).
7.2
A new invariant \(B\), preparation of the free variable (case (ii)).
Let us remind the convention \(d_i(x)=0\) for \( \text{ div}(u_i)\not \subset E,\ 1\le i \le 3\). In particular, \(d_3(x)=0\) in case (ii). For \(B\in \mathbb Q _+\), define the monomial valuation \(v_B\) by
We choose \(B\in \mathbb N \cup \{+\infty \}\) maximal such that, up to the multiplication by an element of \(k(x)^{\times }\), \( \text{ in}_{v_B}(h)\) takes the following form:
with
with \(\lambda _j \in k(x)\) and \((jd_1,jd_2,j\epsilon (x))\in \mathbb N ^3\) whenever equality holds; furthermore, equality holds for some \(j\), \(2 \le j \le m\).
Note that we necessarily have \(\Phi _j(U_1,U_2,U_3)\ne \lambda _j U_3^{j\epsilon (x)}\) for some \(j\) if \(B<+\infty \), since \(B\) is taken to be maximal. Moreover, since \((z,u)\) is fully prepared and \(\mathcal{V}=\langle U_3\rangle \), we necessarily have \(B \ge 1\) and \( \text{ deg}_{U_3}(\Phi _1)< d_3 +\epsilon (x)\).
This construction builds up a face of \(\Delta (h,u,z)\) with equation
for some \(B\) which contains the point \(\mathbf{x}:=(d_1,d_2,\epsilon (x)+d_3)\) and at least another point.
Let \(p\) be the projection
For analytic computations, note that if \(M={z}^{m-j}u_1^{a_1} u_2^{a_2}{u_3}^{a_3}\) is a monomial appearing with nonzero coefficient in some expansion 3.1(1) of \(h\) and \(j\ge 1\), then \(M\) defines the point \(\mathbf{x}_M\):
and
Then \(B+d_1+d_2\) is the minimum value \(x_1+x_2\) for points in \(p(\Delta (h,u,z)\cap \{x_3 < \epsilon (x)+d_3\})\).
We define \(\Delta _2(h;u_1,u_2;u_3)\subseteq (\mathbb R _+)^2\) by the formula
The main idea is that \(\Delta _2(h;u_1,u_2;u_3)\) acts as the characteristic polyhedron of a surface singularity and in the following, we mimic [2, 11], all these following Hironaka.
In case (ii) (\( \text{ div}(u_3) \not \subset E\)), we will require two extra conditions (to be achieved in 7.3 below by possibly changing \(u_3\)):
(P3) there is no homogeneous \(P\in k(x)[U_1,U_2]\), \(P\ne 0\), such that
with the convention \(\lambda _j=0\) when \((jd_1,jd_2,j\epsilon (x))\not \in \mathbb N ^3\);
(P4) if \(B<+\infty \), let \(\mathbf{x}_2=(d_1+A(1),\beta +d_2)\) be the vertex of \(\Delta _2(h;u_1,u_2;u_3)\) with minimal first coordinate. Then \(\mathbf{x}_2\) does not vanish by changing \(u_3\) to \(u_3+\gamma u_1^{\alpha }u_2^{\beta }\), \(\gamma \in R, \gamma \) invertible.
Proposition 7.3
With assumptions as above, there exist \((z,u_{1},u_{2},u_{3}), z,u_3\in \widehat{R}\) such that (P1)(P2) and (P3)(P4) (in case (ii) with \(B<\infty \)) are satisfied.
Proof
The conditions (P1) (P2) can be achieved easily. If (P3) or (P4) is not achieved, we make a translation on \(u_3\): we replace \(u_3\) by \(u_3+\sum _{a_1,a_2} \gamma _{a_1,a_2} u_1^{a_1}u_2^{a_2}\), \(\gamma _{a_1,a_2}\in R\), \((a_1,a_2)\in p(\Delta (h,u,z)\cap \{x_3 < \epsilon (x) + d_3\})\). To achieve (P3), we take
\(P(U_1,U_2)\) as in (36), which makes \(B\) increase if (P3) is not achieved. To achieve (P4) we change \(u_3\) to \(u_3+\gamma u_1^{\alpha }u_2^{\beta }\) as in (P4)), which makes \((A(1),\beta )\) strictly increase for the lexicographical ordering.
In both cases, this translation makes \(\Delta _2(h;u_1,u_2;u_3)\) smaller. These translations may spoil (P1), so each must be followed by a translation on \(z\) to get again (P1). This translation makes \(\Delta _2(h;u_1,u_2;u_3)\) not bigger. The process may be infinite, but since \(\Delta _2(h;u_1,u_2;u_3)\) gets smaller at each step, this converges to some \(z,u_3 \in \widehat{R}\).
Definition 7.3.1
With assumptions as above, a r.s.p. \((z,u_1,u_2,u_3)\), \(z,u_3\in \widehat{R}\) such that (P1)(P2) and (P3)(P4) (in case (ii) with \(B<\infty \)) are satisfied is said to be well prepared. For such \((z,u_1,u_2,u_3)\), the number \(B\) defined above is denoted by \(B(z,u_1,u_2,u_3)\) or \(B(x)\) for short, even if it may depend on the choice of \((z,u_1,u_2,u_3)\).
7.4. We begin the proof of Theorem VI.1 by the special case \(B(x)=\infty \).
When \(B(z,u_1,u_2,u_3)=\infty \), \(\Delta (h,u,z)\) has only one vertex with coordinates \((d_1,d_2,\epsilon (x)+d_3)\). Since \(\epsilon (x)>0\), we have \( \text{ div}(u_3)\not \subset E\), hence \(t(x)=1\) (case (ii)) and \(E\subseteq \text{ div}(u_1u_2)\), \(d_3=0\). The proof is a variation of that of Theorem 3.9, checking carefully the algebraicity of the blowing up centers.
It has been assumed from Sect. 5 on that \( \text{ Sing}_m(X)\subseteq E\), so \(\epsilon (x,E)<1\) necessarily since \(V(z,u_3) \subseteq \text{ Sing}_m(X)\) otherwise.
By blowing up the surfaces \(V(z,u_i)\), \( \text{ div}(u_i)\subseteq E\), it can be assumed w.l.o.g. that \(d_i<1\). Similarly, it can be assumed that \(d_1+d_2<1\) by blowing up \(V(z,u_1,u_2)\).
Assume that \(V(z,u_i,u_3)\subseteq \text{ Sing}_m(X)\subseteq E\), i.e. \(d_i+\epsilon (x)\ge 1\), \(i=1\) or \(i=2\). Then \(C_i:=V(z,u_i,u_3)\) is a formal irreducible component of \( \text{ Sing}_m(X)\). By excellence, its Zariski closure \(\overline{C}_i\) is a curve on \(X\). On the other hand, \(\overline{C}_i\) is contained in \(V(z,u_i)\), so \(C_i\) itself is a curve on \(X\). By blowing up \(C_i\), we may assume that \(d_i +\epsilon (x)<1\), \(i=1,2\).
At this point, we have reached the situation of Theorem 3.9(17) and the proof therein extends without changes: we eventually get reduction in \((m(x),\tau (x))\) by blowing up closed points. We observe that Theorem 7.1 can also be phrased as follows in this case: \(E_n\) can be enlarged to a new normal crossings divisor \(F_n\) such that \(\epsilon (x_n,F_n)=0\).
From now on, we assume that
Definition 7.4.1
(New invariants) We define \(A_1,\beta \) by: \((d_1+A_1,\beta +d_2)\) is the vertex of minimal first coordinate of
We define \(A_2\) by: \(d_2+A_2\) is the minimal second coordinate of the points of
We define \(C(u,z)\) (or \(C(x)\) for short) by;
Finally, we define \(\gamma (u,z)\) (or \(\gamma (x)\) for short) as follows:
-
(i)
\(\gamma (u,z):=\lceil \beta (u,z) \rceil \ge 0\) if (\(E\subseteq \text{ div}(u_1)\) and \(t(x)=1\));
-
(ii)
\(\gamma (u,z):=\lceil \beta (u,z) \rceil \ge 0\) if (\(E\subseteq \text{ div}(u_1u_3)\) and \(t(x)=0\));
-
(iii)
\(\gamma (u,z):=1+\lfloor C(u,z) \rfloor \ge 1\) otherwise, i.e if (\(E= \text{ div}(u_1u_2)\) and \(t(x)=1\) ) or if (\(E= \text{ div}(u_1u_2u_3)\) and \(t(x)=0\)).
Proposition 7.4.2
(Behaviour of the new invariants under blowing up along an \(\epsilon \)-permissible curve). Assume that (Hyp) is true, \((z,u)\) is well prepared and let
Assume that \(\mathcal{C}_i\) is \(\epsilon \)-permissible in \(\widehat{X}= \text{ Spec}(\widehat{R}/(h))\), for some \(i\), \(i=1,2\), then:
-
(i)
\(\epsilon (x)+d_3+d_i \ge 1\),
-
(ii)
\(\mathcal{C}_i\) is algebraic, i.e., if in achieving (P3)(P4), we get \(z,u_3 \in {\widehat{R}}\), then there exists a curve in \(\text{ Spec} R\) whose formal completion is \( \text{ V}(z,u_i,u_3)\).
-
(iii)
let \(\pi _i: (Z^{\prime },x^{\prime }) \rightarrow (Z,x)\) be the blowing up along \(\mathcal{C}_i\), \(X^{\prime } \subset Z^{\prime }\) the strict transform of \(X\) and \(x^{\prime } \in X^{\prime }\) the center of \(v\), with \(\iota (x^{\prime })=\iota (x)\). Then:
-
(iv)
if \(i=1\) and \(x^{\prime }\) is the point of \(Z^{\prime }\) with parameters
$$\begin{aligned} (z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3):=(z/u_1,u_1,u_2,u_3/u_1), \end{aligned}$$these are well-prepared parameters and
$$\begin{aligned}&\beta (x^{\prime })=\beta (x),\ A_1(x^{\prime })=A_1(x)-1,\ A_2(x^{\prime })=A_2(x),\\&d_1(x^{\prime })=d_1(x)+\epsilon (x)+d_3(x)-1, \ d_2(x^{\prime })=d_2(x),\ d_3(x^{\prime })=d_3(x); \end{aligned}$$ -
(v)
if \(i=2\) and \(x^{\prime }\) is the point of \(Z^{\prime }\) with parameters
these are well-prepared parameters and
Proof of (i). Condition (i) is equivalent to \(h\in (z,u_3,u_i)^m\).
Proof of (ii)(iii). Let us note that (ii) is clear when div\((u_3)\subset E\), because in that case, we do not make (P3)(P4), \(z,u_1,u_3 \in R\). When div\((u_3)\not \subset E\), we will prove that
(ii)’ \(\mathcal{C}_i\) is the only analytic branch in \(\text{ div}(u_i)\cap \text{ Sing}_m(X)\cap \{y \in X: \ \epsilon (y)>0 \}\) not contained in div\((u_j)\), \(j=1,2\), \(j\not =i\).
By 3.10, 3.11, this will prove (ii). We compute \(\pi _i:X^{\prime }\subset Z^{\prime } \longrightarrow X^{\prime }\subset Z^{\prime }\). By symmetry, we suppose \(i=1\). Let us expand:
\(C_{A,m-j}\in R\), \(C_{A,m-j}\) invertible or zero, \(C_{A,m-j}\not =0 \Rightarrow a_1+a_2+a_3\ge j\delta (x),\ a_i\ge jd_i,\ i=1,2,3\), \(C_{0,0}\) invertible.
Since \(h \equiv \delta z^m \ \text{ mod}(u_1,u_2,u_3)\), \(\delta \in R\) a unit, \(X^{\prime } \cap \text{ Spec}R[u_1/z,u_3/z]\subset Z^{\prime }\) does not contain the point \((z,u_1/z,u_2,u_3/z)\). Assume now that \(x^{\prime }\) belongs to the affine chart \( \text{ Spec}R[z/u_3,u_1/u_3]\subset Z^{\prime }\). Let
We have
where, \(\gamma _{j}\in R\), \(\gamma _{j}\) invertible or zero, \(\gamma _{j}=0\) when one exponent is not integer,
when \(\gamma _{j}\) is invertible and \(I^+(z^{\prime },u^{\prime }_1,u_2,u_3)\) is generated by
with \(1\le j \le m, a\ge jd_1,b\ge jd_2,c\ge j(d_1+d_3+\epsilon (x)-1), \ a+b+c>j(d_1+d_2+d_1+d_3+\epsilon (x)-1)\).
Note that (38) implies \(d_3(x^{\prime })\ge d_1+d_3+\epsilon (x)-1\), in fact there is equality. Otherwise, by [17], there would exist \(t=z^{\prime }+\gamma u_3^e\), \(e\ge d_1+d_3+\epsilon (x)-1\), \(\gamma \in R\), ord\(_{u_i}\gamma \ge d_i\), \(i=1,2\), with \(h^{\prime }=C_{0,0}t^m \ \text{ modulo}\ I^+(t,u^{\prime }_1,u_2,u_3)\). As \(I^+(z^{\prime },u^{\prime }_1,u_2,u_3)=I^+(t,u^{\prime }_1,u_2,u_3)\) and, in (38), there is no term in \(z^{\prime m-1}\), this is impossible. As \(d_1(x^{\prime })=d_1(x)\) and \(d_2(x^{\prime })=d_2(x)\), by (38), we get \(\epsilon (x^{\prime })=0\): there is no \(x^{\prime }\) very near \(x\) in this chart. This gives the first statement in (iv). This gives also (ii)’, because if there was a curve in div\((u_2)\cap \text{ Sing}_m(X)\cap \{y \in X: \ \epsilon (y) >0 \}\), the strict transform of this curve would have a non empty intersection with our affine chart and there would exist in this chart some \(x^{\prime }\) with \(\epsilon (x^{\prime })\ge 1\).
Proof of (iv). Now \(x^{\prime }\in \text{ Spec}R[z/u_1,u_3/u_1]\subset Z^{\prime }\) is the point with parameters \((z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3):=(z/u_1,u_1,u_2,u_3/u_1)\). Then, using the notations of (37),
\(\Delta (h^{\prime },u^{\prime },z^{\prime })\) is obtained as follows: take the convex hull of the set \(\{(a+c-1,b,c)|(a,b,c)\in \Delta (h,u,z)\}\) and add \(\mathbb{R ^{+}}^{3}\), then
where the \(\lambda _{j,A}\in k(x)\) are defined by:
Let \(\mathcal{M}^{\prime }\) be the set of monomials \(M^{\prime }={z^{\prime }}^{m-j}u_1^{a_1+a_3-j} u_2^{a_2}{u^{\prime }_3}^{a_3}\) which appear with a non zero coefficient in the expansion of \(h^{\prime }\), let \(\mathcal{M}\) be the set of monomials \(M={z}^{m-j}u_1^{a_1} u_2^{a_2}{u_3}^{a_3}\) which appear with a non zero coefficient in the expansion of \(h\):
As \(x^{\prime }\) is very near \(x\), \(\epsilon (x)=\epsilon (x^{\prime })\), \(\delta (x^{\prime })=d_1(x^{\prime })+d_2(x^{\prime })+d_3(x^{\prime })+\epsilon (x)\). The only point on the first side of \(\Delta (h^{\prime },u^{\prime },z^{\prime })\) is
let \(p^{\prime }\) be the projection on \(x_3=0\) from this vertex. A monomial \(M^{\prime }\) defines a point \(({a_1+a_3\over j}-1,{a_2\over j},{a_3\over j})\) that we call also \(M^{\prime }\), when \(a_3<d_3(x)+\epsilon (x)\),
as \({{a_1+a_3\over j}-1-d_1(x^{\prime })\over d_3(x)+\epsilon (x)-{a_3\over j}}= {{a_1\over j}-d_1(x)\over d_3(x)+\epsilon (x)-{a_3\over j}}-1\) and by 7.2 (35)
We get
i.e. the polyhedron \(p^{\prime }(\Delta (h^{\prime },u^{\prime },z^{\prime })\cap \{x_3<d_3(x)+\epsilon (x)\}) \) translated by the vector \(-(d_1(x^{\prime }),d_2(x^{\prime }))\), from
by making an horizontal translation of \(-1\). This gives the other assertions of (iv). Mutatis mutandis, we get (v).
Proposition 7.4.3
(Behaviour of the new invariants under blowing up a closed point) Assume that (Hyp) is true and \((z,u)\) is well prepared. Let \(\pi _i: (Z^{\prime },x^{\prime }) \rightarrow (Z,x)\) be the blowing up along \(x\), \(X^{\prime } \subset Z^{\prime }\) the strict transform of \(X\) and \(x^{\prime } \in X^{\prime }\) the center of \(v\), with \(\iota (x^{\prime })=\iota (x)\). Then
-
(i)
\(x^{\prime }\) belongs to the strict transform of \(V(z,u_3)\),
-
(ii)
if \(x^{\prime }\in \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\) is the point
$$\begin{aligned} (z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3):=(z/u_1,u_1,u_2/u_1,u_3/u_1), \end{aligned}$$these parameters are well-prepared and
$$\begin{aligned}&\beta (x^{\prime })\le \beta (x),\ A_1(x^{\prime })=B(x)-1,\ A_2(x^{\prime })=A_2(x),\ C(u^{\prime },z^{\prime })\le C(u,z), \nonumber \\&d_1(x^{\prime })=d_1(x)+d_2(x)+d_3(x)+\epsilon (x)-1, \ d_2(x^{\prime })=d_2(x),\ d_3(x^{\prime })=d_3(x); \end{aligned}$$ -
(iii)
if \(x^{\prime }\in \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\) and \(x^{\prime } \ne (z^{\prime },u^{\prime }_1,u^{\prime }_2,u^{\prime }_3)\), then
$$\begin{aligned}&\beta (x^{\prime })\le 1 +\left\lfloor {C(u,z)\over 2} \right\rfloor ,\ A_1(x^{\prime })=d(x)-1,\\&d_1(x^{\prime })=d_1(x)+d_2(x)+d_3(x)+\epsilon (x)-1, \ d_2(x^{\prime })=0,\ d_3(x^{\prime })=d_3(x), \end{aligned}$$where \(\lfloor . \rfloor \) denotes lower integral part. If moreover (\(E\subseteq \text{ div}(u_1u_3)\) and \(0<\beta (x)\)), then \(\beta (x^{\prime })\le \beta (x)\). We have
$$\begin{aligned} \gamma (x^{\prime })\le \gamma (x). \end{aligned}$$More precisely: if (\(x^{\prime }\) is not rational over \(x\) and \(\gamma (x)\ge 3\)), then \(\gamma (x^{\prime })<\gamma (x)\); if (\(\gamma (x^{\prime })=\gamma (x)= 2\) and \(x^{\prime }\) is not rational over \(x\)), then \(\beta (x)=2\), \( \text{ div}(u_2)\not \subset E\) and \(\beta (x^{\prime })<\beta (x)=2\);
-
(iv)
if \(x^{\prime }\in \text{ Spec}R[z/u_2,u_1/u_2,u_3/u_2]\subset Z^{\prime }\) is the point with parameters
these are well prepared parameters and
Proof (i) is a consequence of 6.3(ii) and 6.3 (28).
Proof of (ii). Write
where the sum runs along \(b\le m\), \(A=0\) when \(b=0\), and \(A=(a_1,a_2,a_3)\in b\Delta (h,u,z)\). Then
and \(\Delta (h^{\prime },u^{\prime },z^{\prime })\) is obtained as follows: take the convex hull of the set
Let \(\mathcal{M}^{\prime }\) be the set of monomials \(M^{\prime }={z^{\prime }}^{m-j}u_1^{a_1+a_2+a_3-j} {u^{\prime }_2}^{a_2}{u^{\prime }_3}^{a_3}\) which appear with a non zero coefficient in the expansion of \(h^{\prime }\), let \(\mathcal{M}\) be the set of monomials \(M={z}^{m-j}u_1^{a_1} u_2^{a_2}{u_3}^{a_3}\) which appear with a non zero coefficient in the expansion of \(h\):
As \(x^{\prime }\) is very near to \(x\), \(\epsilon (x)=\epsilon (x^{\prime })\), \(\delta (x^{\prime })=d_1(x^{\prime })+d_2(x^{\prime })+d_3(x^{\prime })+\epsilon (x)\). The only point on the first side of \(\Delta (h^{\prime },u^{\prime },z^{\prime })\) is
Let \(p^{\prime }\) be the projection on \(x_3=0\) from this vertex. A monomial \(M^{\prime }\) corresponds to a point \(\mathbf{x}_{M^{\prime }} \ ({a_1+a_2+a_3\over j}-1,{a_2\over j},{a_3\over j})\). When \(a_3<d_3(x)+\epsilon (x)\),
as \({{a_1+a_3\over j}-1-d_1(x^{\prime })\over d_3(x)+\epsilon (x)-{a_3\over j}}= {{a_1\over j}-d_1(x)\over d_3(x)+\epsilon (x)-{a_3\over j}}-1\) and by 7.2 (35)
So we get
from
as follows: take the convex hull of the set
These are the usual transformation laws of the characteristic polyhedra of surfaces see the appendix of H. Hironaka in [3]. To get the other assertions of (ii), the proof runs along the same lines as 7.4.2 (37).
Proof of (iv). Mutatis mutandis, we get all assertions of (iv), except the last line that we prove now. In fact, we get
from
as follows: take the convex hull of the set \(\{(a,a+b-1)+\mathbb{R _+}^2|(a,b)\in p(\Delta (h,u,z)\cap \{x_3<d_3(x)+\epsilon (x)\})-(d_1(x),d_2(x))\}\). We get \(A_1(x^{\prime })=A_1(x)\), \(\beta (x^{\prime })=\beta (x)+A_1(x)-1\) and \(A_2(x)=d(x)-1\).
Let us denote by \((\alpha _2,\beta _2)\) and \((\alpha _3,\beta _3)\) with \(\alpha _2\le \alpha _3\), the coordinates of the (maybe equal) vertices of the first side of \(p(\Delta (h,u,z)\cap \{x_3<d_3(x)+\epsilon (x)\})-(d_1(x),d_2(x))\).
Then \((\alpha _2,\alpha _2+\beta _2-1)=(\alpha _2,B(x)-1)\) is the vertex of smaller second coordinate of
Note that \((A_1(x),A_1(x)+\beta (x)-1)\) is the vertex of smaller first coordinate of
All this leads to:
Then either \(B(x)-A_1(x)\le {\beta (x)\over 2}\), then \(C(u^{\prime },z^{\prime })\le {\beta (x)\over 2}\) by the last inequality; or \(B(x)-A_1(x)> {\beta (x)\over 2}\), then \(C(u^{\prime },z^{\prime })< {\beta (x)\over 2}\) by the first of the two inequalities just above. The inequality \(\gamma (x^{\prime })\le \gamma (x)\) is left to the reader.
Proof of (iii). Recall the notations and assumptions of 7.2 (34). We write
with \(\lambda _j \in k(x), \lambda _j =0\) if \((jd_1,jd_2,j\epsilon (x))\not \in \mathbb N ^3\). In this expansion, we take:
Note that at least one \(Q_{i,j}, 2\le j \le m\) is nonzero and at least one \(\lambda _{j^{\prime }}\), \(2\le j^{\prime } \le m\) is nonzero.
By definition of \(C(u,z)\), when \(Q_{i,j}\not =0\), deg\((Q_{i,j})\le iC(u,z)\), where deg is the usual homogeneous degree. When \(Q_{i,j}\not =0\), let us denote \(d(i,j)=\)deg\((Q_{i,j})\). Then we have, with natural notations, the relation:
which leads to:
Then, in the expansion of \(U_3^{j\epsilon (x)+jd_3-i}U_1^{a(i,j)}U_2^{b(i,j)}Q_{i,j}(U_1,U_2)\), the monomial with non zero coefficient and minimal exponent in \(U_1\) is
which gives the point (cf. 7.2 (35))
in \(p(\Delta (h,u,z)\cap \{x_3<\epsilon (x)+d_3\})\). As
we deduce that
Similarly,
and, finally, by (40), when \(Q_{i,j}\not =0\),
Since \(x^{\prime } \in \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\), \(x^{\prime }\) is not the origin
and \(x^{\prime }\) belongs to the strict transform of V\((z,u_3)\) then \(z^{\prime }(x^{\prime })=u^{\prime }_1(x^{\prime })=u^{\prime }_3(x^{\prime })=0\). We complete \((z^{\prime },u^{\prime }_1,u^{\prime }_3)\) to a r.s.p. \((z^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3)\) at \(x^{\prime }\) where
for some irreducible polynomial
The following lemma will end the proof of 7.4.3(iii).
Lemma 7.4.4
With hypotheses and notations as in 7.4.3(iii), let \(d:=[k(x^{\prime }):k(x)]\). We have:
-
(i)
\(A_1(x^{\prime })=B(u,z)-1\);
-
(ii)
if \( \text{ div}(u_3)\subset E\), then
$$\begin{aligned} \beta (x^{\prime })\le {C(u,z) \over d }\le {\beta (x) \over d } \end{aligned}$$ -
(iii)
in general,
$$\begin{aligned} \beta (x^{\prime })< 1+\left\lfloor {C(u,z) \over d } \right\rfloor , \end{aligned}$$(44) -
(iv)
if (\(E \subseteq \text{ div}(u_1)\), \(0< \beta (x)\) and \(x^{\prime }\) is rational over \(x\)), then
Proof.
As \(x^{\prime }\) is very near to \(x\), we have \(\epsilon (x)=\epsilon (x^{\prime })\), \(\delta (x^{\prime })=d_1(x^{\prime })+d_2(x^{\prime })+d_3(x^{\prime })+\epsilon (x)\). As \(x^{\prime }\) is on the strict transform of div\((u_3)\) and not on the strict transform of div\((u_2)\), we get:
With notations as in the proof of 7.4.3(ii):
where the sum runs along \(b\le m\), \(A=0\) when \(b=0\), \(A=(a_1,a_2,a_3)\in b\Delta (h,u,z)\),
Up to multiplying \(h\) by an unit, we may assume \(\overline{C_{0,m}}=1\in k(x)\). Then, with the notations of 3.2, we have
\(\mu _j=0\) whenever \((jd_1,jd_2,jd_3+j\epsilon (x)) \not \in \mathbb N ^3\), \(\mu _j=\overline{C_{jd_1,jd_2,jd_3+j\epsilon (x),m-j}}\in k(x)\) otherwise. This leads to
where \(h^{\prime }_1 \in \{{z^{\prime }}^{m-j} {u^{\prime }_1}^{a(j)}, j\in \mathbb N , a(j)>j(d_1+d_2+d_3+\epsilon (x)-1) =j(\delta (x)-1)\}\). As a consequence,
is the vertex of smallest first coordinate of \(\Delta (h^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3,t)\) and is not solvable. In the preparation, we may replace \(z^{\prime }\) by \(t=z^{\prime }+\lambda {u^{\prime }_1}^a\) with \(a\ge \delta (h)-1\), but, this cannot erase the vertex \((\delta (x)-1,0,d_3(x)+\epsilon (x))\). We get
Let us study the projection of \(\Delta (h^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3,t)\cap \{x^{\prime }_3<d_3+\epsilon (x^{\prime })\}\) on \(x^{\prime }_3=0\), in particular we are interested in the vertex of smallest first coordinate of this projection. Let \(w\) be the monomial valuation on \(R^{\prime }:=\mathcal{O}_{X^{\prime },x^{\prime }}\) defined by
There is an expansion
where
where \(d(i,j)= \text{ deg}(Q_{i,j}) \overline{u^{\prime }_2}\) is the image of \(u^{\prime }_2\) in \({R^{\prime } \over (z^{\prime },u_1,u^{\prime }_3)} = k(x)[\overline{u^{\prime }_2}]_{(\overline{v^{\prime }})}\), \(\overline{v^{\prime }}\) being the image of \(v^{\prime }\) in \({R^{\prime } \over (z^{\prime },u_1,u^{\prime }_3)}\). Let us recall that
with \(I_r=\{a\in R^{\prime }|w(a)\ge r\}\), \(I_r^+=\{a\in R^{\prime }|w(a)> r\}\). By 7.4.3 (40), \(\text{ in}_{w}(h^{\prime })=\)
This means that
is the defining equation of a face of \(\Delta (h^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3)\).
7.4.4.1 When div\((u_3)\subset E\), then div\((u^{\prime }_3)\subset E^{\prime }\), we have just to make (P2) in the preparation, we may replace \(z^{\prime }\) by \(t=z^{\prime }+r\), \(r\in R^{\prime }\) and, as \({Z^{\prime }}^{m-1}\) does not appear in (2), \(w(r)>1\), \(w(t)=w(z^{\prime })=1\). This means that
is the defining equation of a face of \(\Delta (h^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3,t)\). By 7.2 (34),
By 7.4.3(5), \(\beta (x^{\prime })\le C(u,z)/d\) and this gives 7.4.4 in the case div\((u_3)\subset E\).
7.4.4.2 From now on, \( \text{ div}(u_3)\not \subset E\), in particular \(d_3(x)=0\). Then, to get (P1),\(\ldots \),(P4), we may replace \(z^{\prime }\) by \(t=z^{\prime }+r\), \(r\in R^{\prime }\) and, as \({Z^{\prime }}^{m(x)-1}\) does not appear in (46), \(w(r)>1\), \(w(t)=w(z^{\prime })=1\). We possibly have to make the projection of \(\Delta (h^{\prime },u^{\prime }_1,v^{\prime },u^{\prime }_3,t)\cap \{x^{\prime }_3<d_3+\epsilon (x^{\prime })\}\) on \(x^{\prime }_3=0\) smaller by changing \(u^{\prime }_3\) to \(v_3=u^{\prime }_3 +\lambda {u^{\prime }_1}^a\) with \(a\ge B(u,z)-1\) and \(\lambda \in R^{\prime }\), \(\lambda \) not divisible by \(u^{\prime }_1\).
Assume that \(a>B(u,z)-1\) (this is always the case when \(B(u,z)\not \in \mathbb N \)). Then in\(_w(v_3)=\)in\(_w(u^{\prime }_3)\), we get \(A_1(u^{\prime }_1,v^{\prime },u^{\prime }_3,t)=B(u,z)-1\) and
By 7.4.3 (43),
which gives 7.4.4 in this case.
7.4.4.3 From now on,
If there exists a couple \((i,j_0)\) such that in (47) above
then the translations \(t=z^{\prime }+r\) and \(v_3=u^{\prime }_3 +\lambda {u^{\prime }_1}^a\) will not modify the term
with \(i_0:= \text{ min}\{i : Q_{i,j_0}\not =0\}\). More precisely, in the expansion
\(\mu _{i,j}\in {R^{\prime } \over (t,u^{\prime }_1,u^{\prime }_3)}, e(i,j)\in \mathbb N \), we will have
Then
which, by 7.4.3 (43), gives
and implies 7.4.4 in this case.
7.4.4.4 From now on, we assume the implication:
In particular, we have \(j\epsilon (x)\in \mathbb N \), \(j\delta (x)\in \mathbb N \) and all the indices \(i\) in (46) (47) are integers. Let us define
by
by
so (47) can be rewritten:
The preceding remarks rewrite \(j\epsilon (x)\not \in \mathbb N \Rightarrow F_j=0, F^{\prime }_j=0\). Let
deg\(_{U_3}(G_j)=m!\epsilon (x)\) or \(G_j=0\), and deg\(_{U^{\prime }_3}(G^{\prime }_j)=m!\epsilon (x)\) or \(G^{\prime }_j=0\).
Let \(\mu _1, \mu _2 \in k(x)\), \(j_1,j_2 ,\ 2\le j_1,j_2 \le m\), let
where \(Q_i=0\) or \(Q_i\) neither divisible by \(U_1\) nor by \(U_2\). Let us denote \(d(i):=\,\)deg\((Q_i)\). Assume that for some \(i\), \(Q_i\not =0\), then, by 7.4.3 (43),
Assume that not all \(G_j\)’s are collinear in the \(k(x)\)-vector space \( \text{ gr}_{v_B}(R)\). Then there is some \(G\not =0\) as above with \(\lambda =0\). Let
with some \(Q_i\not =0\). Let \(i_0:= \text{ min}\{i: Q_{i_0}\not =0\}\). Let
Replacing \(U^{\prime }_3\) by \(V_3\), we get
Then
which implies 7.4.4 in this case.
7.4.4.5 From now on, we assume that all \(G_j\)’s are collinear in the \(k(x)\)-vector space \( \text{ gr}_{v_B}(R)\).
By (P3) for \((z,u)\), any \(G_j\not =0\) is not collinear to a \((m!\epsilon (x))^\mathrm{th}\)-power, any \(F_j\not =0\) is not collinear to a \((j\epsilon (x))^\mathrm{th}\)-power. Take some \(F_j\not =0\), and let
Let \(v_3=u^{\prime }_3 +\lambda {u^{\prime }_1}^a\), with \(a\ge B(u,z)-1\) and \(\lambda \in R^{\prime }\), \(\lambda \) not divisible by \(u^{\prime }_1\). Let
When \(b<\beta _0\), we have
When \(b>\beta _0\), we have
When \(b=\beta _0\) and there exists \(i<p^e\) such that \(Q_{i,j}\not =0\), say \(i_0\) is the smallest such \(i\), we get
When \(b=\beta _0\) and for \(i\le p^e\) \(Q_{i,j}=0\), then
When \(b=\beta _0\, \text{ and} \text{ for}\, i< p^e Q_{i,j}=0 \text{ and} \lambda _j^{-1}U_1^{a(p^e,j)} U_2^{b(p^e,j)}Q_{p^e,j} \text{ is} \text{ a} (p^e)\)th-power, then
with \(a\ge A_1(x), b\ge A_2(x) \text{ and} B_0:= \text{ deg}(Q_0)\le C(u,z)\). Let \(W_3:=U_3+U_1^{a}U_2^{b}Q_0\), we get
Let \(w_3\in R\) such that in\(_{v_B}(w_3)=W_3\), then, with \(w^{\prime }_3=w_3/u_1\), \(W^{\prime }_3=\)in\(_w(w^{\prime }_3)\):
Let \(v_3=w^{\prime }_3+\lambda ^{\prime } {u^{\prime }_1}^{a^{\prime }}\), with \(a^{\prime }\ge B(u,z)-1\) and \(\lambda ^{\prime } \in R^{\prime }\), \(\lambda ^{\prime }\) not divisible by \(u^{\prime }_1\). Then we conclude as above:
7.4.4.6 From now on, we assume 7.4.4.2, 7.4.4.3, 7.4.4.4, 7.4.4.5 and for \(i< p^e\) \(Q_{i,j}=0\) and \({\lambda _j}^{-1}U_1^{a(p^e,j)} U_2^{b(p^e,j)}Q_{p^e,j}\) is NOT a \((p^e)^\mathrm{th}\)-power (c.f. (49)). In particular \(e\ge 1\). Let us recall the following elementary lemma [13, II.5.3.2].
Lemma 7.4.4.7
Let \(F(U_1,U_2)\in k(x)[U_1,U_2]\) be a homogeneous polynomial of degree \(d_0\ge 0\), and \(a,b\in \mathbb N \) be such that \(U_1^aU_2^b F(U_1,U_2)\not \in (k(x)[U_1,U_2])^p\).
Let \(x^{\prime }\in \text{ Spec}k(x)[{U_2 \over U_1}]\) be a closed point with ideal \((v:=P(1, {U_2 \over U_1}))\), \(P\in k(x)[U_1,U_2]\) a nonzero homogeneous irreducible polynomial of degree \(d:=[k(x^{\prime }):k(x)]\), unitary in \(U_2\).
Let \(A\in T^{\prime }:=k(x)[U_1,{U_2 \over U_1}]_{(U_1,v)}\) be such that \(U_1^{a+b+d_0}\) (resp. \(U_1^{a+b+d_0}v^b\)) divides \(A^p\) in \(T^{\prime }\) if \(P\ne U_2\) (resp. \(P=U_2\)). There exists an integer \(c \ge 0\) such that
with \(\gamma \) invertible in \(T^{\prime }\). We have the following estimates for \(c\):
-
(i)
if \(P\ne U_2\) (resp. \(P=U_2\)), then \(c\le 1+{d_0\over d}\) (resp. \(c\le d_0\));
-
(ii)
if \(P \ne U_2\), then \(c<p(1+ \lfloor {d_0\over pd}\rfloor )\) (equivalently: for every \(N\in \mathbb N \) such that \({d_0 \over pd}<N\), we have \(c<Np\));
-
(iii)
if \(d_0\ge 1\) and \(b=0\), then \(c \le i\).
Let \(f<e\) be the integer defined by:
Let
with \(p^{h}a=a(p^e,j)\), \(p^{h}b=b(p^e,j)\), \(p^{h}d_0=d(p^e,j)\), where \(d_0:=\)deg\((Q_{0})\). In particular,
Then,
By Lemma 7.4.4.7, \((\overline{u^{\prime }_2}^{b(i_0,j)}Q_{i_0,j}(1,\overline{u^{\prime }_2})+ \overline{\lambda }^{p^e})=(\gamma {v^{\prime }}^c)^{(p^{h})} \not =(0)\), so
Furthermore, by (i) above, \(c\le 1+{d_0 \over d}\), so:
By (ii),
which is 7.4.4(iii). Now 7.4.4(iv) is a consequence of (iii) above.
7.4.5 Proof of Theorem VI.1: some cases with \(\gamma (u,z)=1\).
The strategy to make the proof is to make a list of different subcases covering this case, from the easiest to the most difficult and to prove them up to the former ones.
All cases ( \(\beta (u,z)<1\) and \( \text{ div}(u_1u_2) \not \subset E\)) are covered by 7.4.5.3 below. All cases with (\(\gamma (u,z)<1\) and \( \text{ div}(u_1u_2) \subseteq E\)) are dealt with in 7.4.5.6. This includes in particular all remaining cases with \(\beta (u,z)<1\) since \(C(u,z) \le \beta (u,z)\) for \( \text{ div}(u_1u_2) \subseteq E\) (see Definition 7.4.1).
Lemma 7.4.5.1
With assumptions as in 7.4.3, assume furthermore that
There exist well prepared parameters \((z^{\prime },u^{\prime })\) at \(x^{\prime }\) such that
Proof
This is a direct consequence of Lemma 7.4.4.
Lemma 7.4.5.2
With assumptions as in 7.4.3, assume furthermore that
Then \(\mathcal{C}_1:=V(z,u_1,u_3)\) is an \(\epsilon \)-permissible algebraic curve on \(X\).
Let \(\pi : (Z^{\prime },x^{\prime }) \rightarrow (Z,x)\) be the blowing up along \(\mathcal{C}_1\), \(X^{\prime } \subset Z^{\prime }\) the strict transform of \(X\) and \(x^{\prime } \in X^{\prime }\) the center of \(v\) and assume \(\iota (x^{\prime })=\iota (x)\). Then \((z^{\prime },u^{\prime })=(z/u_1,u_1,u_2,u_3/u_1)\) are well prepared parameters at \(x^{\prime }\) and we have
Proof
This follows from Proposition 7.4.2.
Remark 7.4.5.3
Lemmas 7.4.5.1 7.4.5.2 prove Theorem 7.1 when (Hyp) is true and
Indeed, \(d_3(x)+\epsilon (x)=d_3(x^{\prime })+\epsilon (x^{\prime })\) if \(\iota (x^{\prime })=\iota (x)\) after blowing up. If \(E\subseteq \text{ div}(u_1u_3)\), then \(E^{\prime }\subseteq \text{ div}(u^{\prime }_1u^{\prime }_3)\), so condition (3) remains stable after blowing up. A descending induction on \(A_1(u,z)\) ends the proof.
7.4.5.4 Proof of Theorem 7.1 in the case \(C(u,z)=0\).
In that special case, we have \(\beta (u,z)=A_2(u,z)\). When \(A_2(u,z)<1\) and \(A_1(u,z)<1\), 7.4.5.1 gives the result. Let us see the other cases:
Case 1 \(d_3(x)+\epsilon (x)\ge 1\). We may assume \(A_2(u,z)\ge 1\) by symmetry on \(u_1,u_2\). Then \(\mathcal{C}:=(z,u_2,u_3)\) is an \(\epsilon \)-permissible algebraic curve by Proposition 7.4.2 and we get
after blowing up along \(\mathcal{C}\) if \(\iota (x^{\prime })=\iota (x)\). A descending induction on \(A_2(x)\) and 7.4.5.2 give the result. From now on, we assume
Case 2 \(d_3(x)+d_i(x)+\epsilon (x)<1\), \(i=1\) and \(i=2\). We blow up \(\{x\}\) in this case. By Proposition 7.4.3(ii) or (iv), we have
if \(\iota (x^{\prime })=\iota (x)\) and \(x^{\prime }\) is the origin of a chart. Otherwise, Lemma 7.4.4(ii)(iii) gives 7.4.5.2(52) at \(x^{\prime }\) for some well prepared r.s.p. \((z^{\prime },u^{\prime })\).
Case 3 \(d_3(x)+\epsilon (x)< 1\), \(d_3(x)+d_i(x)+\epsilon (x)\ge 1\) for some \(i=1\) or \(i=2\), \(A_j(x)\ge 1 \), \(j=1,2\). We choose an \(\epsilon \)-permissible blowing up center \(Y\) as follows:
if V\((z,u_i,u_3)\), for \(i=1,2\) are \(\epsilon \)-permissible, then \(Y:=(z,u_i,u_3)\) with
if V\((z,u_i,u_3)\) is \(\epsilon \)-permissible for a unique \(i \in \{1,2\}\), then \(Y:=(z,u_i,u_3)\);
if V\((z,u_i,u_3)\) is not \(\epsilon \)-permissible for \(i \in \{1,2\}\), then \(Y:=\{x\}\).
Let \(n(x):=2\) if \((A_1(x),d_1(x))=(A_2(x),d_2(x))\), \(n(x):=1\) otherwise. If \(\iota (x^{\prime })=\iota (x)\), we claim that \(x^{\prime }\) satisfies 7.4.5.2 (52) or falls into cases 1,2 above, or there is a well prepared r.s.p. \((z^{\prime },u^{\prime })\) at \(x^{\prime }\) with \(C(u^{\prime },z^{\prime })=0\) and
Note that this ends the proof of the case \(C(u,z)=0\), since (3) can repeat but finitely many times. To prove the claim, first assume that \(Y=(z,u_1,u_3)\). By Proposition 7.4.2, we have
and the result is clear. The case \(Y=(z,u_2,u_3)\) is similar.
Assume now that \(Y=\{x\}\). By symmetry on \(u_1,u_2\), we assume \(A_2(x)\ge 1\). If \(x^{\prime }\in \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\) and \(x^{\prime }\) is the point with parameters \((z/u_1,u_1,u_2/u_1,u_3/u_1)\) (origin of the first chart), we get
Since \(A_2(x)\ge 1\) and \(V(z,u_2,u_3)\) is not \(\epsilon \)-permissible, we have
We get \(d_1(x)>d_2(x)\) and \( d_1(x^{\prime })=d_1(x)+d_2(x)+d_3(x)\epsilon (x)-1<d_1(x)\), \(A_1(x^{\prime })<A_1(x)\le A_2(x^{\prime })=A_2(x)\) which proves the claim.
If \(x^{\prime }\in \text{ Spec}R[z/u_1,u_2/u_1,u_3/u_1]\subset Z^{\prime }\) and \(x^{\prime }\) is not the above point, we have 7.4.5.2 (52) at \(x^{\prime }\) for some well prepared r.s.p. \((z^{\prime },u^{\prime })\) at \(x^{\prime }\) by lemma 7.4.4(ii)(iii).
If \(x^{\prime }\in \text{ Spec}R[z/u_2,u_1/u_2,u_3/u_2]\subset Z^{\prime }\) and \(x^{\prime }\) is the point with parameters \((z/u_2,u_1/u_2,u_2,u_3/u_2)\) (origin of the second chart), we get
We have \(A_1(x) < 1\): otherwise, as \(V(z,u_1,u_3)\) is not \(\epsilon \)-permissible, this would imply \(d_1(x)+d_3(x)+\epsilon (x)<1\), \(d_2(x)+d_3(x)+\epsilon (x)\ge 1\), hence \(V(z,u_2,u_3)\) \(\epsilon \)-permissible since \(A_2\ge 1\): a contradiction. We now get \(A_1(x)<1\le A_2(x)\) and \(A_2(x^{\prime })=A_1(x)+A_2(x)-1<A_2(x)\) which completes the proof of the claim.
7.4.5.5 Proof of Theorem 7.1 in the case \(C(u,z)<1\), \( \text{ div}(u_1u_2)\subseteq E\).
We perform the sequence of local blowing ups
where \(x_0=x\), \(x_i\in X_i\) (\(X_i\) denoting the strict transform of \(X\)), \(0\le i \le n\), is the center of \(v\), each blowing up center is \(Y_i=\{x_i\}\).
If \(\iota (x_1)=\iota (x)\), and is not the origin of a chart (viz. case 3 in 7.4.5.4), then \(x_1\) verifies the assumptions of 7.4.5.1 by Lemma 7.4.4(iii).
Assume now that \(\iota (x_i)=\iota (x)\) and \(x_i\) is the origin of a chart for all \(i\ge 0\). By 7.4.3(ii)(iv), \(x_i\) verifies the assumptions of 7.4.5.5 and \(C(x_{i+1})\le C(x_i)\) for all \(i\ge 0\). It is then a very well known fact that \(C(x_i)=0\) for \(i>>0\), i.e. the assumptions of 7.4.5.4 are satisfied.
7.4.5.6 End of the proof of Theorem 7.1. As our invariants \(C(u,z)\), \(\beta (u,z)\) are discrete, the next lemma shows that we will reach one of the cases (ii) \(\beta (u,z)<1\) or (iii) \(C(u,z) <1\). This ends the proof of Theorem 7.1 (see comments right after 7.4.5.).
Lemma 7.4.5.7
With assumptions as in 7.4.3, consider the sequence of local blowing ups
where \(x_0=x\), \(x_i\in X_i\) (\(X_i\) denoting the strict transform of \(X\)), \(0\le i \le n\), is the center of \(v\), each blowing up center is \(Y_i=\{x_i\}\).
Assume that \(\iota (x_i)=\iota (x)\) for all \(i \ge 0\). There exists some \(i \ge 0\) and a well prepared r.s.p. \((z_i,u_{1,i},u_{2,i},u_{3,i})\) at \(x_i\) (w.r.t. the reduced inverse image of \(E\) in \(Z_i\)) such that one of the following holds:
-
(i)
\(\gamma (u_i,z_i)<\gamma (u,z)\);
-
(ii)
\(\beta (u_i,z_i)<1\);
-
(iii)
\(C(u_i,z_i)<1\);
Proof This breaks up in three cases:
Case 1 for all \(i\ge 0\), the point \(x_i\) is the origin of one of the two charts of 7.4.3, i.e. we are always in one of the cases 7.4.3(ii)(iv). Then \(C(u_i,z_i)=0\) for \(i>>0\) (see 7.4.5.5 above).
Case 2 for all \(i \ge 0\), (\(x_i\) is rational over \(x\) and belongs to the first chart), i.e. \(x_i\) is a rational point not on the strict transform of \( \text{ div}(u_1)\). By 7.4.3(ii)(iii), \(x_1\) has a r.s.p. of the form \((z/u_1,u_1,u_2/u_1+\mu _1,u_3/u_1)\) for some \(\mu _1\in R\). A well prepared r.s.p. is of the form
with \(\mu _2 =0\) if \( \text{ div}(u_3) \subseteq E\). Then there exists a regular formal curve \(\mathcal{C}\) of the form \(\mathcal{C}=V(\widehat{z},\widehat{u_2},\widehat{u_3})\) on \( \text{ Spec}(\widehat{R}/(h)\), transverse to \(E_i\) for all \(i\ge 0\), \(\widehat{u_3}=u_3\) if \( \text{ div}(u_3) \subseteq E\), whose strict transform goes through all points \(x_i\), \(i \ge 0\). Necessarily \(\mathcal{C} \subseteq \text{ Sing}_m(X)\), so we may assume that \(\mathcal{C}\subset \text{ div}(u_j) \subseteq E\) for \(j=2\) or \(j=3\). In particular, we may take \(\widehat{u_j}=u_j\) for \(j=2\) or \(j=3\). This implies that \(v(u_j)>v(u_1^n)=nv(u_1)\) for all \(n\ge 1\): a contradiction, since our given valuation \(v\) has rank one.
Case 3. \(E\subseteq \mathrm{div}(u_1u_3)\) and we are not in case 2, i.e. there exists \(i_0\ge 0\) such that either \(x_{i_0+1}\) is not rational over \(x_{i_0}\) or \(E_{i_0+1}\) has one more component than \(E_{i_0})\), \(i_0\) minimal. Suppose \(\beta (x_{i_0})\ge 1\).
If \(x_{i_0+1}\) is not rational over \(x_{i_0}\), we get
by 7.4.3(ii)(iii) and 7.4.4(iii): note that \(C(u,z)\le \beta (u,z)\) since \(\text{ div}(u_2) \not \subset E\) and
If \(E_{i_0+1}\) has one more component than \(E_{i_0}\), we have
by 7.4.3(ii), (iii), (iv). This gives 7.4.5.7(iii) if \(1 \le \beta (x)<2\).
Now, \(\gamma (x_{i_0+1})= 1+\lfloor C(u_{i_0+1},z_{i_0+1})\rfloor \), \(\gamma (x)=\lceil \beta (x)\rceil \), so we get 7.4.5.7(i) if \(\beta (x)>2\).
Assume that
Since \(\gamma (x)=2\), we get 7.4.5.7(i) unless \(\gamma (x_i)=2\) for \(i\ge 0\) by Proposition 7.4.3.
Let \(i_1>i_0\) be the largest index such that \(E_i\) has as many components as \(E_{i_0}\) for \(i_0 \le i \le i_1\). We may assume \(i_1<+\infty \) by case 1 and we have
By 7.4.4(iii), we get \(\beta (x_{i_1+1})<2\). Now the point \(x_{i_1+1}\) falls into case 2 above or into case 3 with (56) not satisfied. This concludes the proof in case 3.
The end of the proof of 7.4.5.7 is just a logical game: we reach the assumption \(E \subseteq \text{ div}(u_1u_3)\) for some point \(x_i\), \(i\ge 0\) provided we are not in case 1.
References
Cossart, V.: Sur le polyèdre caractéristique d’une singularité. Bull. Soc. Math. F 103, 13–19 (1975)
Cossart, V.: Desingularization of embedded excellent surfaces. Tohoku Math. J. II. Ser 33, 25–33 (1981)
Cossart, V.: Resolution of surface singularities. In: Lecture Notes in Mathematics, vol. 1101, pp. 79–98. Springer, Berlin (1984)
Cossart, V.: Forme normale d’une fonction sur un k-schéma de dimension 3 and de caractéristique positive. Géométrie algébrique et applications, C. R. 2ieme Conf. int., La Rabida/Espagne, 1984: I: Géométrie and calcul algébrique. Trav. Cours 22, 1–21 (1987)
Cossart, V.: Sur le polyèdre caractéristique. Thèse d’État Orsay, pp. 1–424 (1987)
Cossart, V.: Polyèdre caractéristique et éclatements combinatoires. Rev. Mat. Iberoam. 5(1/2), 67–95 (1989)
Cossart, V.: Contact maximal en caractéristique positive and petite multiplicité. Duke Math. J. 63(1), 57–64 (1991)
Cossart, V.: Modèle projectif régulier et désingularisation. Math. Ann. 293(1), 115–122 (1992)
Cossart, V.: Désingularisation en dimension 3 et caractéristique it p. Proceedings de La Rabida. Progress in mathematics, vol. 134, pp. 1–7. Birkhauser, Boston (1996)
Cossart, V.: Uniformisation et désingularisation des surfaces. dédié à O. Zariski (Hauser, Lipman, Oort, Quiros Éd.). Progress in mathematics, vol 181, pp. 239–258. Birkhauser, Boston (2000)
Cossart, V., Janssen, U., Saito, S.: Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes. arXiv:0905.2191
Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. J. Algebra 320(3), 1051–1082 (2008)
Cossart, V., Piltant, O.: Resolution of singularities of threefolds in positive characteristic. II. J. Algebra 321(7), 1836–1976 (2009)
Giraud, J.: Étude locale des singularités Cours de 3‘eme cycle, Pub. no 26. Univ. d’Orsay (1972)
Giraud, J.: Contact maximal en caractéristique positive. Ann. Scient. Ec Norm. Sup. 4ème série, t.8, pp. 201–234 (1975)
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic 0, I–II. Ann Math. 109–326 (1964)
Hironaka, H.: Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7(3), 251–293 (1967)
Hironaka, H.: Additive groups associated with points of a projective space. Ann. Math 92, 327–334 (1970)
Hironaka, H.: Idealistic exponents of singularity. (J.J. Sylvester symposium, John Hopkins University, Baltimore, 1976). John Hopkins University Press, pp. 52–125 (1977)
Hironaka, H.: Theory of infinitely near singular points. J. Korean Math. Soc. 40(5), 901–920 (2003)
Matsumura, H.: Commutative ring theory. In: Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cossart, V., Piltant, O. Resolution of singularities of threefolds in mixed characteristic: case of small multiplicity. RACSAM 108, 113–151 (2014). https://doi.org/10.1007/s13398-012-0103-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-012-0103-5