RESOLUTION OF SINGULARITIES OF THREEFOLDS IN POSITIVE CHARACTERISTIC A fiew obstructions to embedded resolution.

Vincent COSSART

INTRODUCTION

The purpose of this conference is to give some hints about the proof of the theorem cited below and to show on two examples that the notion of maximal contect fails completely in the case of the positive characteristic. Theorem. (Cossart et Piltant) [CP1,CP2]. Let k be a field of positive characteristic which is differentially finite over a perfect field k_{0} and Z/k be a reduced quasiprojective scheme of dimension three with singular locus Σ. There exists a projective morphism π : Z̃ → Z, such that

(i) Z̃ is regular. (ii) π induces an isomorphim Z̃\π^{−1}(Σ) ≃ Z\Σ. (iii) π^{−1}(Σ) ⊂ Z̃ is a divisor with strict normal crossings.

I Main reduction.

In [CP1], we prove that the proof may be reduced to the proof of this theorem below.

Theorem. Let k be a field of positive characteristic which is differentially finite over a perfect field k_{0}, i.e. Ω_{k/k_{0}}^{1} has finite dimension.

Let S be a regular local ring of dimension three, essentially of finite type over k and such that K := QF (S) has transcendance degree 3 over k. Let R̄ be an Artin-Schreier or purely inseparable singularity of dimension three over S.

Let K := QF (S) and L := QF (R̄) (in particular L/K is a finite field extension).

Then, each k-valuation μ of L dominating R̄ and satisfying properties (i) and (ii) below has a local uniformization:

(i) μ has rank one and κ(μ)/κ(S) is algebraic; (ii) μ is the unique extension of its restriction to K.

I.1 Notations.

We let R := S[X]_{(X,m_{S})}, X_{0} =Spec(R/(h), x_{0} his closed point, 𝔐 = (X, m_{S}), and k(x_{0}) = R/𝔐 is a finitely generated field extension of k. We denote by (u_{1}, u_{2}, u_{3}) a regular system of parameters (r.s.p. for short) of S, so 𝔐 = (X, u_{1}, u_{2}, u_{3}). We assume all along this text that h is irreducible over S[X], i.e. that f is not of the form θ^{p} − θg^{p−1} for any θ ∈ S.

If g ≠ 0, such a singularity is called “Artin-Schreier”, if g = 0, it is called “purely inseparable”.

I.2 Purely inseparable case.

Let us concentrate on the purely inseparable case which already contains enough difficulties. To simplify, we take

h = X^{p} + f(u_{1}, u_{2}, u_{3}), f ∈ k[u_{1}, u_{2}, u_{3}].

We suppose that there is an exceptional divisor E which contains locally the singular locus of h = 0 and such that I(E) divises u_{1}u_{2}u_{3}. This can be achieved easily (see [CP1]). We suppose that ord_{x_{0}}(f) ⩾ p (else, the singularity is eqasily solved). We define J (f, E), the ideal generated by the coefficients of df ∈ Ω_{S/k_{0}}^{1}(logE). We defined H(x) = ∏_{div(u_{i})⊂E} U_{i}^{a(i)}, where a(i) =ord_{u_{i}}(J (f, E)), then set

J(f, E) := H(x)^{−1}J (f, E).

The main invariant is:

ω(x) := ord_{x}(J(f, E))

which, obviously does not depend on X. I.2.1 Important remark. The case ω(x_{0}) = 0 is easily solved: see [CP2] II.4.4 to II.4.7.

It can be shown that

ω(x_{0}) = 0 ⇔ f = γMmodS^{p}, M = ∏_{div(u_{i})⊂E} u_{i}^{a(i)},

with γ invertible or parameter orthogonal to E and, if γ invertible, then M is not a p-power.

Villamayor’s example. In his conference, O. Villamayor prosed the following example as a “terminal case”.

X^{2} + u^{2}v, p = 2, E = div(u).

The computations give:

J (E) = (u^{2}), H(x) = u^{2}, J(E) = (1), ω(x_{0}) = 0.

We agree with Orlando: it is a terminal case.

All this leads to:

I.2.2 Second reduction.

Let W := SpecS, find W′, some iterated blowing up of W, where x ∈ W′ the center of the restriction of μ in W′ verifies:

ω(x) < ω(x_{0}).

The problem is reduced to a problem of moniomalization modulo p-powers in a smooth variety: the problem is more difficult, but the dimension of the ambient space drops of one.

I.2.3 Permissible centers.

Now we forget X_{0} and we work in W, E is a normal crossing divisor in W. We do not write here the definitions [CP2, chapter 1, II.5] of the permissible centers. We just recall that, if Y is permissible at x, them it is smooth at x and locally normal crossing with E, furthermore, closed points are permissible. Bad news: permissible centers are not necessarily contained in the locus where ω is maximal. We have the following propositions.

I.2.3.1 Proposition. If Y is irreducible of generic point ξ and permissible at x, then ω(ξ) ⩾ ω(x) − 1.

I.2.3.2 Transformation laws. Let Y be as above and π : X′ ⟶ X′ be the blowing up centred at Y. Then J(f′, E′) is the weak transform of J(f, E, Y) (defined below), where f′ is the strict transform of f and E′ the total transform of E.

I.2.3.3 Jacobian adapted to Y.

Let

D(E) := {D ∈ Der_{k_{0}}O_{W} | D(I(E)) ⊂ I(E)},
D(E, Y) := {D ∈ D(E) | D(I(Y)) ⊂ I(Y)}.

Then

J(f, E, Y) := H(x)^{−1}(D(f), D ∈ D(E, Y).

I.2.3.4 Example.

In the case where Y = x and E=div(u_{1}u_{2}), k_{0} = k, we get

D(E, Y) = (\frac{u_{1}∂}{∂u_{1}} , \frac{u_{2}∂}{∂u_{2}} , 𝔐\frac{∂}{∂u_{3}} ).

I.2.3.5 Proposition. With notations as above, if x′ ∈ W′ is above x, then

ω(x′) ⩽ ω(x).

II No maximal contact for ω.

It is very well known that there is no maximal contact for the Hilbert-samuel function in characteristic p > 0 (see section III).

Optimist people may think that there may be maximal contact for the couple (HS, ω) where HS is the Hilbert-Samuel function. This is wrong. The following example shows that there cannot exist in X a surface Σ such that the Hilbert-Samuel function the strict transforms of Σ is constant at the points x_{i} above x with (HS(x_{0}), ω(x_{0})) = (HS(x_{i}), ω(x_{i})), here HS(x) is just the local multiplicity which is equal to p.

II.1. Example in dimension 3. (joint work with O. Piltant).

With the notations of I.2. We have a singularity in dimension 3 of equation:

h = X^{p} + u_{1}^{a}u_{2}^{b}(v^{p^{e}} + (u_{2} − u_{1})^{p^{e}}u_{1}^{p+2} + extra).

extra ∈ S = k[u_{1}, u_{2}, v], of very big order, E = div(u_{1}u_{2}), a + b = 0 mod p, ab ≠ 0 mod p, p ≠ 2. We suppose k = k_{0} algebraically closed.

Computations give:

H(x_{0}) = u_{1}^{a}u_{2}^{b}, J(E) = (v^{p^{e}}) mod 𝔐^{1+p^{e}}, ω(x_{o}) = p^{e}.

We blow up along the origin.

We take as new origin the point x_{1} of parameters

(X/u_{1}, u_{1}, w := (u_{2} − u_{1})/u_{1}, v/u_{1})

that we note (X, u_{1}, w, v), using an usual convention.

We get

h_{1} = X^{p} + u_{1}^{a+b−p}(w + 1)^{b}(v^{p^{e}} + w^{p^{e}}u_{1}^{p+2} + u_{1}^{A}extra′)

where h_{1} is the strict transform of h, A ∈ ℕ, A very big. The reader sees that the set of points x′ above x_{0} with (HS(x_{0}), ω(x_{0})) = (HS(x′), ω(x′)) is exactly Proj(X, V), where V :=in_{x_{0}}(v), this implies that, if Σ exists, its directrix [CJS,1.20] at x_{0} is V(X, V).

We change one variable: let Y := X + u1^{\frac{a+b−p}{p}}, then
h_{1} = Y^{p} + u_{1}^{a+b−p}(γv^{p^{e}}w + w^{p^{e}}u_{1}^{p+2} + u_{1}^{A}extra′),

E_{1} = div(u_{1}), h(x_{1}) = u_{1}^{a+b−p}, J(E_{1}) = (v^{p^{e}}, w^{p^{e}}u_{1}^{p+2}) mod (u_{1}^{A}), ω(x′) = p^{e}.

We go on: we look at the sequences of blowing ups centered at closed points on the strict transform of (Y, v, w). We use the usual convention i.e. we note the parameters of x_{i+1} (Y, u_{1}, v, w) instead of (Y/u_{1}^{i}, u_{1}, v/u_{1}^{i}, w/u_{1}^{i}).

We get at x_{2}:

h_{2} = X^{p} + u_{1}^{something}(w + 1)^{b}(v^{p^{e}} + w^{p^{e}}u_{1}p + 1 + u_{1}^{A}extra_{2})

E_{2} = div(u_{1}), h(x_{2}) = u_{1}^{something}, J(E_{2}) = (v^{p^{e}}, w^{p^{e}}u_{1}^{p+1}) mod (u_{1}^{A}), ω(x′) = p^{e}.

We get at x_{p+3}:

h_{p+3} = X^{p} + u_{1}^{something}(w + 1)^{b}(v^{p^{e}} + w^{p^{e}} + u_{1}^{A}extra_{3})
E_{p+3}div(u_{1}), h(x_{3}) = u_{1}^{something}, J(E_{3}) = (v^{p^{e}}, w^{p^{e}}) mod (u_{1}^{A}), ω(x′) = p^{e}.

Things are looking well: the initial part of J(E_{p+3}) is (v^{p^{e}}, w^{p^{e}}), the directrix of the ideal J(E_{p+3}) has dimension 1: the equations are v = w = 0, the dimension was 2 for J(E_{i}), i < p + 3, the equation was v = 0.

We get at x_{p+4}:

h_{p+4} = X^{p} + u_{1}^{something}(w + 1)^{b}(v^{p^{e}}u_{1} + w^{p^{e}} + u_{1}^{A}extra_{4})

E_{p+4} = div(u_{1}), h(x_{4}) = u_{1}^{something}, J(E_{4}) = (v^{p^{e}}u_{1}, w^{p^{e}}) mod (u_{1}^{A}), ω(x′) = p^{e}.

The initial part of J(E_{p+4}) is (w^{p^{e}}), the directrix of the ideal J(E_{p+4}) has dimension 2: the equation is w = 0.

We get at x_{p+3+N}:

h_{p+3+N} = X^{p} + u_{1}^{something}(w + 1)^{b}(v^{p^{e}}u_{1}^{N} + w^{p^{e}} + u_{1}^{A}extra_{p+3+N})

E_{p+3+N} = div(u_{1}), h(x_{p+3+N}) = u_{1}^{something}, J(E_{p+3+N}) = (v^{p^{e}}u_{1}^{N}, w^{p^{e}}) mod (u_{1}^{A}), ω(x′) = p^{e}.

The initial part of J(E_{p+3+N}) is (w^{p^{e}}), the directrix of the ideal J(E_{p+3+N}) has dimension 2: the equation is w = 0.

Let us blow up along x_{p+3+N}: it is easily seen that the set of points x′ above x_{p+3+N} with (HS(x_{0}), ω(x_{0})) = (HS(x′), ω(x′)) is exactly Proj(X, W), where W :=in_{x_{p+3+N}} (w), this implies that, if Σ exists, the directrix of its strict transform Σ_{p+3+N} at x_{p+3+N} is V(X, W). This is impossible, by [CJS,12.1,12.3,13.3], its ideal should be (X, V) mod (U_{1}).

II.2 Conclusion

This kills the hope to have a maximal contact in “Giraud’s sense” for (HS, ω), i.e. to find a surface Σ such that the strict drop of its local Hilbert-Samuel function would imply a strict drop of some invariant of the original singularity.

The question is: define another invariant, finer than (HS, ω) such that, for this invariant, you get a maximal contact in “Giraud’s sense”... For the moment, there is no answer.

III No maximal contact along a valuation. (joint work with U. Jannsen and S. Saito).

III.1. Recall of the algorithm, dimension 2, hypersurface case.

We follow Hironaka in [CGO, appendix]. X is a singular surface embedded in a 3-dimensional smooth k-variety Z, k is an algebraic closed field of characteristic p > 0. We suppose that the worse HS-stratum is a finite union of closed points, in this case the worse HS-stratum is the locus of maximal multiplicity μ(x), we call it HS(X)

The algorithm is: blow up the locus of multiplicity ⩾ μ(x). This will stop.

More precisely, in an open neighbourhood U ⊂ X of x ∈ HS(X), you blow up X, Z along x:

X_{1} ⊂ Z_{1} ⟶ X ⊂ Z.

One can show that there are three different cases [CJS, section 2].

(i) Either there is no point in X_{1} near to x (no point with same multiplicity): STOP, the maximum multiplicity dropped strictly above x. (ii) Either there are a finite number of closed points in X_{1} near to x, then above U, blow up X_{1}, Z_{1} along these points:

X_{2} ⊂ Z_{2} ⟶ X_{1} ⊂ Z_{1}.

(iii) Or the set of points in X_{1} near to x is a projective line D_{1} then above U, blow up X_{1}, Z_{1} along D_{1}:

X_{2} ⊂ Z_{2} ⟶ X_{1} ⊂ Z_{1},

either there is no point in X_{2} near to x, either there are a finite number of closed points in X_{2} near to x or the set of points in X_{2} near to x is a projective line D_{2} which projects isomorphically on D_{1}, then above U, blow up X_{2}, Z_{2} along D_{2}.

In the latter case, above U, the algorithm creates a “fundamental sequence” [CJS, section 5], i.e. a finite sequence of blowing ups

X_{m} ⊂ Z_{m} ⟶ X_{m−1} ⊂ Zm − 1 ⟶ ⋯ X_{2} ⊂ Z_{2} ⟶ X_{1} ⊂ Z_{1} ⟶ X ⊂ Z,

where the center of the blowing up X_{i} ⊂ Z_{i} ⟶ X_{i−1} ⊂ Z_{i−1} is a projective line D_{i−1} which projects isomorphically on D_{1}, 2 ⩽ i ⩽ m and either there is no point in X_{m} near to x, or there are a finite number of closed points in X_{m} near to x.

III.2 Maximal contact along a valuation

The example given in [CJS, section 15] shows that there is no maximal contact in positive characteristic. What is new in this paper is that there is no maximal contact along a valuation. Let us recall the definition of maximal contact [CJS, section 15]. III.2.1 Definition: hypersurface of maximal contact.

Let Z be an excellent regular scheme and X ⊂ Z be a closed subscheme.

A closed subscheme W ⊂ Z is called to have maximal contact with X at x ∈ X if the following conditions are satisfied:

(i) x ∈ W. (ii) Take any sequence of permissible blowups [...]:

\begin{aligned}Z = & Z_{0} & \underleftarrow{π_{1}} & Z_{1} & \underleftarrow{π_{2}} & Z_{2} & ← … ← & Z_{n−1} & \underleftarrow{π_{n}} & Z_{n} & ← ⋯ \\ & ∪ & & ∪ & & ∪ & & ∪ & & ∪ \\ X = & X_{0} & \underleftarrow{π_{1}} & X_{1} & \underleftarrow{π_{2}} & X_{2} & ← … ← & X_{n−1} & \underleftarrow{π_{n}} & X_{n} & ← ⋯ \end{aligned}  (1)

where for any n ⩾ 0

\begin{aligned}Z_{n+1} & = & Bl_{D_{n}}(Z_{n}) & \underleftarrow{π_{n+1}} & Z_{n} \\ & & ∪ & & ∪ \\ X_{n+1} & = & Bl_{D_{n}}(X_{n}) & \underleftarrow{π_{n+1}} & X_{n} \end{aligned}  (2)

and D_{n} ⊂ X_{n} is permissible. Assume that there exists a sequence of points x_{n} ∈ D_{n} (n = 0, 1, …) such that x_{0} = x and x_{n} is near to x_{n−1} for all n ⩾ 1. Then D_{n} ⊂ W_{n} for all n ⩾ 0, where W_{n} is strict transform W in Z_{n}.

Some optimist people asked us:

“Your definition of maximal contact is weaker than Hironaka’s, but still too strong, for the uniformization problem, you just need the definition below. Have you an example where there is no maximal contact along a valuation?”

We found one just before Kyoto conference. III.2.2 Definition: maximal contact along a valuation. In the definition above suppose X is a projective variety over a field k, let ν be a k-valuation, then on each X_{n}, ν has a center x_{n}.

We say that a closed subscheme W ⊂ Z has maximal contact with X along ν if, for every sequence (1) where x_{0} = x and x_{n} is near to x_{n−1} for all n ⩾ 1, then x_{n} ∈ W_{n} for all n ⩾ 0, where W_{n} is strict transform W in Z_{n}.

Indeed if for any k-valuation ν there could exist a smooth W (ν) satisfying III.2.2, life would be much easier (as says the guru Abhyankar) in desingularization theory. III.2.3 The example.

p = 3

No exeptional divisor E, E = ∅, take

f := y^{3} + u_{2}^{2}[(u_{1}^{3} − u_{2}^{2})(u_{1}^{3} + u_{2}^{2})^{3} + u_{1}^{N}], N ≠ 0 mod (p), N >> 0.

Let us recall some definitions. III.2.3.1 Hironaka’s polyhedrons. [H1] or [CJS, section 7] In the case where

f = y^{m} + ∑_{i,a,b,a+b>i,0⩽i⩽m} λ_{i,a,b}y^{m−i}u_{1}^{a}u_{2}^{b}, λ_{i,a,b} ∈ k,

Δ(f, u, y) is the convex hull spanned by \{(\frac{a}{i}, \frac{b}{i}), λ_{i,a,b} ≠ 0\} + ℝ_{⩾0}^{2}. Hironaka defines Δ(f, u) as:

Δ(f, u) = ∩_{y,in_{𝔐}(f)=Y^{m}}Δ(f, u, y).

III.2.3.2 Notations. δ(f, u, t) =inf{a + b, (a, b) ∈ Δ(f, u, t)}, δ(f, u) =inf{a + b, (a, b) ∈ Δ(f, u)}.

We write sometimes δ(x) instead of δ(f, u) where x is the point of parameters (y, u_{1}, u_{2}), indeed, one can prove that δ(f, u) does not depend on (u_{1}, u_{2}). In our example, we get

δ(x) = 3 + 1/3

III.2.3.3 ϵ(x). With the notations and hypotheses of III.2.3.1, assume that there is an exceptional divisor E with components smooth and orthogonal to y = 0, then assume that

E ⊂ div(u_{1}u_{2}),

(we say E has “new components”), we define A_{1} =inf{a|(a, b) ∈ Δ(f, u)}, mutatis mutandis, we define A_{2}. Then

ϵ(x) := δ(x) − ∑_{div(u_{i})⊂E} A_{i}.

In our example, E = ∅, we get

ϵ(x) = δ(x) = 3 + 1/3

III.2.3.4 Change of ϵ(x). In the example, to avoid useless denominators, we will replace ϵ(x) by 3ϵ(x) that we still denote by ϵ(x). From now on:

ϵ(x) = 10.

III.2.3.5. In the example, x is isolated in the HS-stratum.

Indeed

\frac{∂f}{∂u_{1}} = u_{1}^{N−1}u_{2}^{2}.

So, if a curve is contained in the Hs-stratum at the beginning, u_{1}^{N−1}u_{2}^{2} has order at least 2 along this curve which is contained in div(u_{1}) or div(u_{2})

(i) if it is contained in div(u_{2}), as f = y^{3} mod (u_{2}), the only possibility is the curve V(y, u_{2}) which does not fit (ii) if it is contained in div(u_{1}), it should be V(z, u_{1}) with y^{3} + u_{2}^{10} ∈ (z, u_{1})^{3}

\frac{∂y^{3} + u_{2}^{10}}{∂u_{1}} = u_{2}^{9} ∈ (z, u_{1})^{2}

so z = u_{2}, which does not fit.

III.2.3.6 Let us start Hironaka’s algorithm. We blow up along the origin and take the point x_{1} above of parameters

y/u_{1}, u_{1}, u_{2}/u_{1} ϵ(x_{1}) = 6

y^{(1)}^{3} + u_{1}^{7}u_{2}^{(1)}^{2}[(u_{1} − u_{2}^{(1)}^{2})(u_{1} + u_{2}^{(1)}^{2})^{3} + u_{1}^{N−8}]

The exceptional divisor is div(u_{1}) “new component”. Following Hironaka, we make the “fundamental sequence”: we get the point x_{3} above of parameters

(y^{(3)}, u_{1}^{(3)}, u_{2}^{(3)}) := (y^{(1)}/u_{1}^{2}, u_{1}, u_{2}^{(1)}),

f_{3} := y^{(3)}^{3} + u_{1}^{(3)}u_{2}^{(3)}^{2}[(u_{1}^{(3)} − u_{2}^{(3)}^{2})(u_{1}^{(3)} + u_{2}^{(3)}^{2})^{3} + u_{1}^{(3)}^{N−8}] ϵ(x_{3}) = 6

We make again the “fundamental sequence”: we first blow up along x_{3}, we look at the point x_{4} above of parameters

(y^{(4)}, u_{1}^{(4)}, u_{2}^{(4)}) := (y^{(3)}/u_{2}^{(3)}, u_{1}^{(3)}/u_{2}^{(3)}, u_{2}^{(3)})

f_{4} := y^{(4)}^{3} + u_{1}^{(4)}u_{2}^{(4)}^{4}[(u_{2}^{(4)} − u_{2}^{(4)})(u_{2}^{(4)} + u_{2}^{(4)})^{3} + u_{1}^{(4)}^{N−8}u_{2}^{(4)}^{N−13}], ϵ(x_{4}) = 4

div(u_{1}^{(4)}u_{2}^{(4)}) is the exceptional divisor, both components are “new”: we end this second “fundamental sequence” we look at the point x_{5} above of parameters

(y^{(5)}, u_{1}^{(5)}, u_{2}^{(5)}) := (y^{(4)}/u_{2}^{(4)}, u_{1}^{(4)}, u_{2}^{(4)})

f_{5} := y^{(5)}^{3} + u_{1}^{(5)}u_{2}^{(5)}[(u_{1}^{(5)} − u_{2}^{(5)})(u_{1}^{(5)} + u_{2}^{(5)})^{3} + u_{1}^{(5)}^{N−8}u_{2}^{(5)}^{N−12}], ϵ(x_{5}) = 4.

Following the algorithm, we blow up along the origin, above at the point of parameters z := y^{(5)}/u_{1}^{(5)}, v_{1} := u_{1}^{(5)}, v := u_{2}^{(5)}/u_{1}^{(5)} + 1, let us see that the ϵ-invariant increases strictly:

ϵ(x_{6}) = 5 (kangaroo point as defined by H. Hauser [HH]).

III.2.3.7 Surprising computation.

Exercise for the reader: compute ω(x_{5}) and ω(x_{6}) (cf. I.2). The main point is that z^{3} +v_{1}^{3}(v − 1)[(v + 1)v^{3} + v_{1}^{2N−24}(v − 1)^{N−12}] and (2, 1) a solvable vertex of Δ(f, z, v_{1}, v).

Let us solve it.

f = z^{3} + v_{1}^{3}[(v^{2} − 1)v^{3} + v_{1}^{N−12}(v − 1)^{N−11}]
w := z − v_{1}^{2}v.

This gives

f = w^{3} + v_{1}^{3}[v^{5} + v_{1}^{N−12}(v − 1)^{N−11}]

As N ≠ 0 mod (3), Δ(f, w, v_{1}, v) has two non solvable vertices (non integer coordinates)

(1, 5/3), (N/3 − 4, 0),
div(v_{1}) is the new component: ϵ(x_{6}) = 5 > ϵ(x_{5}) = 4.

III.2.3.8 No maximal contact on this example.

I claim that, in this example, if we end the fundamental sequence at x_{6} and add another fundamental sequence, there will be a point x_{9} ∈ X_{9} such that there exists no t = y − γ, γ ∈ k[[u_{1}, u_{2}]] such that the x_{i} are on the strict transform of div(t), 0 ⩽ i ⩽ 8. One can see that it implies that there is no smooth hypersurface W ⊂ Z such that the x_{i} are on the strict transform of W ⊂ Z. We define x_{7} as the point on the strict transform of div(v) in the bu along x_{6}. These points x, x_{1},....,x_{8},x_{9} are near to each other: there is a valuation ν of the function field whose center on X_{i} is x_{i}, 0 ⩽ i ⩽ 9: there is no maximal contact ALONG THE VALUATION ν. So no hope of maximal contact, even for the uniformization problem or Hironaka’s game.

Suppose a smooth hypersurface W ⊂ Z has maximal contact along ν, let us call t = 0 its equation in a neighbourhood of x.

III.2.4 Proposition [CJS, section 15] Suppose x = x_{0} isolated in its HS-stratum, then if there exists a smooth hypersurface t = 0 such that along a fundamental sequence starting at x = x_{0} the x_{j}, j ⩾ 0 (resp. j ⩾ i) are on the strict transform of div(t), then

t = y − γ, γ ∈ k[[u_{1}, u_{2}]], ⌊δ(f, u, t)⌋ = ⌊δ(f, u)⌋.

As δ(x) = 3 + 1/3, by III.2.4, γ ∈ (u_{1}, u_{2})^{3}.

γ = P_{3}(u_{1}, u_{2}) + P_{4}(u_{1}, u_{2}) + ρ, ρ ∈ (u_{1}, u_{2})^{5},

P_{i}(u_{1}, u_{2}) ∈ k[u_{1}, u_{2}], homogeneous of degree i or = 0, i = 3, 4.

f := t^{3} + u_{2}^{2}[(u_{1}^{3} − u_{2}^{2})(u_{1}^{3} + u_{2}^{2})^{3} + u_{1}^{N}] + P_{3}^{3} + P_{4}^{3} + ρ^{3}.

f_{1} = t^{(1)}^{3} + u_{1}^{7}u_{2}^{(1)}^{2}[(u_{1} − u_{2}^{(1)}^{2})(u_{1} + u_{2}^{(1)}^{2})^{3} + u_{1}^{N−8}] + u_{1}^{9}P_{3}(1, u_{2}^{(1)})^{3} + u_{1}^{9}P_{4}(1, u_{2}^{(1)})^{3} + u_{1}^{12}ρ′.

f_{3} := t^{(3)}^{3} + u_{1}^{(3)}u_{2}^{(3)}^{2}[(u_{1}^{(3)} − u_{2}^{(3)}^{2})(u_{1}^{(3)} + u_{2}^{(3)}^{2})^{3} + u_{1}^{(3)}^{N−8}] + P_{3}(1, u_{2}^{(3)})^{3} + u_{1}^{(3)}^{3}P_{4}(1, u_{2}^{(3)})^{3} +

u_{1}^{(3)}^{6}ρ′^{3},

δ(x_{3}) = 2 + 1/3, so by III.2.4, P_{3}(1, u_{2}^{(3)}) ∈ (u_{2}^{(3)})^{2}, let us denote P_{3}(1, u_{2}^{(3)}) = au_{2}^{(3)}^{2} + bu_{2}^{(3)}^{3}, a, b ∈ k.

f_{4} := t^{(4)}^{3} + u_{1}^{(4)}u_{2}^{(4)}^{4}[(u_{2}^{(4)} − u_{2}^{(4)})(u_{2}^{(4)} + u_{2}^{(4)})^{3} + u_{2}^{(4)}^{N−8}u_{2}^{(4)}^{N−13}] + a^{3}u_{2}^{(4)}^{3} + b^{3}u_{2}^{(4)}^{6} +

u_{1}^{(4)}^{3}P_{4}(1, u_{2}^{(4)})^{3} + u_{1}^{(4)}^{6}u_{2}^{(4)}^{3}ρ′′^{3}.

As V(y^{(4)}, u_{2}^{(4)}) is permissible, it is contained in div(t^{(4)}), so V(y^{(4)}, u_{2}^{(4)}) =V(t^{(4)}, u_{2}^{(4)})

P_{4}(1, u_{2}^{(4)}) = λu_{2}^{(4)} + cu_{2}^{(4)}^{2} + du_{2}^{(4)}^{3} + eu_{2}^{(4)}^{4}, λ, c, d, e ∈ k.

f_{5} := t^{(5)}^{3} + u_{1}^{(5)}u_{2}^{(5)}[(u_{1}^{(5)} − u_{2}^{(5)})(u_{1}^{(5)} + u_{2}^{(5)})^{3} + u_{1}^{(5)}^{N−8}u_{2}^{(5)}^{N−12}] + a^{3} + b^{3}u_{2}^{(5)}^{3} + u_{1}^{(5)}^{3}(λ^{3} +

c^{3}u_{2}^{(5)}^{3} + d^{3}u_{2}^{(5)}^{6} + e^{3}u_{2}^{(5)}^{9}) + u_{1}^{(5)}^{9}ρ′′^{3}, δ(f_{5}, u) = 2, by III.2.4,

a = b = λ = 0, ρ′ not invertible.

f_{7} := t^{(6)}^{3} + v_{1}^{3}(v − 1)[(v + 1)v^{3} + v_{1}^{N−12}(v − 1)^{N−12}] + v_{1}^{3}(c^{3}(v − 1)^{3} + d^{3}(v − 1)^{6}v_{1}^{3} + e^{3}(v − 1)^{9}v_{1}^{6}) + v_{1}^{6} × something

f_{7} = w^{3} + v_{1}^{3}[v^{5} + v_{1}^{N−12}(v − 1)^{N−11}].

δ(x_{7}) = 2 + 2/3.

We end the fundamental sequence, we get

f_{8} = t^{3} + (v − 1)[(v + 1)v^{3} + v_{1}^{N−12}(v − 1)^{N−12}] + c^{3}(v − 1)^{3} + d^{3}(v − 1)^{6}v_{1}^{3} + e^{3}(v − 1)^{9}v_{1}^{6} + v_{1}^{3} × something

f_{8} = w^{3} + v^{5} + v_{1}^{N−12}(v − 1)^{N−11}

δ(x_{8}) = 5/3, c = 0, else there is no point on the strict transform of div(t) and x_{8} is, furthermore by III.2.4,

d = 0.

We go on: we blow up x_{8} and we look at the near point x_{9} on the strict transform of v = 0.

f_{9} = w^{3} + u^{2}v^{5} + u^{N−15}(uv − 1)^{N−11}

As x_{9} is supposed to be on the strict transform of div(t),

something is not invertible.

f_{9} = t^{3} +(uv−1)[(uv+1)v^{3} +u^{N−15}(v−1)^{N−12}]+e^{3}(uv−1)^{9}u^{3} +u^{3} ×something′, something′ is invertible or divisible by u^{3}.

In any case, as δ(x_{8}) = 7/3 > 2, the monomial (uv − 1)(uv + 1)v^{3} gives a contradiction with III.2.4.

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