S INGULARITIES IN POSITIVE CHARACTERISTIC

S INGULARITIES IN POSITIVE CHARACTERISTIC

A. Benito, A. Bravo and O. Villamayor U.

Universidad Autónoma de Madrid

RIMS Kyoto, December 2008

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Multiplicity and theτ-invariant

k[x ₁ , . . . , x _n ]

g(x ₁ , . . . , x _n ) = G _b (x ₁ , . . . , x _n )+G _b+1 (x ₁ , . . . , x _n )+· · ·+G _N (x ₁ , . . . , x _n ) G _b 6= 0 ⇐⇒ ν x (g) = b at a local ring of the origin.

Spec(k [x ₁ , . . . , x _n ]), X = {g = 0} ⊃ F _b set of b − fold points.

X X ₁ X s

V ←− V ₁ ←− . . . ←− V _s

F _b F _b ¹ F _b ^s

(1)

Problem: Given F _b ⊂ X ⊂ V find a sequence as (1) so that F _b ^s = ∅.

k[x ₁ , . . . , x _n ]

g(x ₁ , . . . , x _n ) = G _b (x ₁ , . . . , x _n )+G _b+1 (x ₁ , . . . , x _n )+· · ·+G _N (x ₁ , . . . , x _n ) G _b 6= 0 ⇐⇒ ν x (g) = b at a local ring of the origin.

Spec(k [x ₁ , . . . , x _n ]), X = {g = 0} ⊃ F _b set of b − fold points.

X X ₁ X s

V ←− V ₁ ←− . . . ←− V _s

F _b F _b ¹ F _b ^s

(1)

Problem: Given F _b ⊂ X ⊂ V find a sequence as (1) so that F _b ^s = ∅.

Multiplicity and theτ-invariant

k[x ₁ , . . . , x _n ]

g(x ₁ , . . . , x _n ) = G _b (x ₁ , . . . , x _n )+G _b+1 (x ₁ , . . . , x _n )+· · ·+G _N (x ₁ , . . . , x _n ) G _b 6= 0 ⇐⇒ ν x (g) = b at a local ring of the origin.

Spec(k [x ₁ , . . . , x _n ]), X = {g = 0} ⊃ F _b set of b − fold points.

X X ₁ X s

V ←− V ₁ ←− . . . ←− V _s

F _b F _b ¹ F _b ^s

(1)

Problem: Given F _b ⊂ X ⊂ V find a sequence as (1) so that F _b ^s = ∅.

k[x ₁ , . . . , x _n ]

g(x ₁ , . . . , x _n ) = G _b (x ₁ , . . . , x _n )+G _b+1 (x ₁ , . . . , x _n )+· · ·+G _N (x ₁ , . . . , x _n ) G _b 6= 0 ⇐⇒ ν x (g) = b at a local ring of the origin.

Spec(k [x ₁ , . . . , x _n ]), X = {g = 0} ⊃ F _b set of b − fold points.

X X ₁ X s

V ←− V ₁ ←− . . . ←− V _s

F _b F _b ¹ F _b ^s

(1)

Problem: Given F _b ⊂ X ⊂ V find a sequence as (1) so that F _b ^s = ∅.

Multiplicity and theτ-invariant

V ^(d) smooth scheme of dimension d over k X ⊂ V ^(d) hypersurface, b the highest multiplicity of X

F _b := {x ∈ X | mult _x (X ) = b}

x ∈ F _b if and only ν x (I(X )) = b

Notation: ν x denotes the order in the local regular ring O _V

,x

Fix X , I(X ) ⊂ O _V

,x

{x ₁ , . . . , x _d } r.s.p.

gr _M

(O _V

) ∼ = k ⁰ [X ₁ , . . . , X _d ] I X ,x initial ideal of I(X ) in gr _M

(O _V

)

τ x : least number of variables needed to express a generator of I X,x .

Notation: τ X = τ

Local presentation

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

Local presentation

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

Local presentation

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

Local presentation

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν _x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν _x (β ^∗ (I ^(s) )) ≥ s}

L OCAL PRESENTATION AND PERMISSIBLE TRANSFORMATIONS

Y ⊂ F _b permissible center, Y ∼ = β(Y )

X X ₁

Y ⊂ (V ^(d) , x )

β

(V ₁ ^(d) , x ⁰ )

oo π

β

β(Y ) ⊂ (V ^(d−τ) , x _τ ) ^{oo e π} (V ₁ ^(d−τ) , x _τ ⁰ ) I ^(s) O

V

= I(H ₁ ^(d−τ) ) ^s · I ₁ ^(s) ,

where H ₁ ^(d−τ) is the exceptional locus of e π

Local presentation

Since

β : (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) the blow-up induces

β ₁ : (V ₁ ^(d) , x ⁰ ) −→ (V ₁ ^(d−1) , x ₁ ⁰ ) −→ . . . −→ (V ₁ ^(d−τ) , x _τ ⁰ )

(1)’ Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ as before.

(2)’ Monic polynomials, g _i ^(p

⁾ for i = 1, . . . , τ (the strict transforms of the monic polynomials f _i ^(p

⁾ ).

(3)’ The ideal I ₁ ^(s) in O

V

with the same positive integer s.

(1)’+(2)’+(3)’ define a local presentation.

Since

β : (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) the blow-up induces

β ₁ : (V ₁ ^(d) , x ⁰ ) −→ (V ₁ ^(d−1) , x ₁ ⁰ ) −→ . . . −→ (V ₁ ^(d−τ) , x _τ ⁰ )

(1)’ Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ as before.

(2)’ Monic polynomials, g _i ^(p

⁾ for i = 1, . . . , τ (the strict transforms of the monic polynomials f _i ^(p

⁾ ).

(3)’ The ideal I ₁ ^(s) in O

V

with the same positive integer s.

(1)’+(2)’+(3)’ define a local presentation.

Local presentation

A sequence of permissible transformations

X X ₁ X r

V ^{(d) oo π}

V ₁ ^(d) oo . . . ^{oo π}

V _r ^(d) induces

V ^(d)

β

V ₁ ^(d)

π

oo

β

. . .

oo V ^(d) _r ^π

^oo

β

V ^{(d−τ) oo e π}

V ₁ ^(d−τ) oo . . . ^{oo e π}

V _r ^(d−τ) and

V ^{(d−τ) oo} V ₁ ^(d−τ) oo . . . ^oo V _r ^(d−τ)

I ^(s) I ₁ ^(s) I _r ^(s)

A sequence of permissible transformations

X X ₁ X r

V ^{(d) oo π}

V ₁ ^(d) oo . . . ^{oo π}

V _r ^(d) induces

V ^(d)

β

V ₁ ^(d)

π

oo

β

. . .

oo V ^(d) _r ^π

^oo

β

V ^{(d−τ) oo e π}

V ₁ ^(d−τ) oo . . . ^{oo e π}

V _r ^(d−τ) and

V ^{(d−τ) oo} V ₁ ^(d−τ) oo . . . ^oo V _r ^(d−τ)

I ^(s) I ₁ ^(s) I _r ^(s)

Local presentation

A sequence of permissible transformations

X X ₁ X r

V ^{(d) oo π}

V ₁ ^(d) oo . . . ^{oo π}

V _r ^(d) induces

V ^(d)

β

V ₁ ^(d)

π

oo

β

. . .

oo V ^(d) _r ^π

^oo

β

V ^{(d−τ) oo e π}

V ₁ ^(d−τ) oo . . . ^{oo e π}

V _r ^(d−τ) and

V ^{(d−τ) oo} V ₁ ^(d−τ) oo . . . ^oo V _r ^(d−τ)

I ^(s) I ₁ ^(s) I _r ^(s)

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Strategy in characteristic zero

S TRATEGY IN CHARACTERISTIC ZERO

A local presentation of X (at x ) when char (k) = 0 is (1) Positive integers 0 = e ₁ = e ₂ = · · · = e _τ . (2) Regular parameters

f ₁ ⁽¹⁾ (z ₁ ) = z ₁ ∈ O _V

[z ₁ ] .. .

f _τ ⁽¹⁾ (z _τ ) = z _τ ∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

Stage A): Reduction to the monomial case.

X X ₁ X _r

V ^{(d) oo} V ₁ ^(d) oo . . . ^oo V _r ^(d) is defined so that, setting

V ^{(d−τ )} ←− V ₁ ^(d−τ) ←− . . . ←− V _r ^(d−τ) and I ^(s) as above, then

I _r ^(s) = I(H ₁ ^(d−τ) ) ^α

· I(H ₂ ^(d−τ) ) ^α

. . . I(H _r ^(d−τ) ) ^α

⊂ O

V

.

Strategy in characteristic zero

Stage B): Resolution of the monomial case.

X r X _r+1 X _N

V _r ^{(d) oo} V _r+1 ^(d) oo . . . ^oo V _N ^(d) is defined so that setting

V _r ^{(d−τ) oo} V _r ^(d−τ) ₊₁ oo . . . ^oo V _N ^(d−τ) I _r ^(s) I _r+1 ^(s) I _N ^(s) and I _i ^(s) ⊂ O

V

as before, then

{x ∈ V _N ^(d−τ) | ν x (I _N ^(s) ) ≥ s} = ∅ (easy).

Key Point

The local presentation in char zero will allow us to lift sequence V _r ^{(d−τ) oo} V _r ^(d−τ) ₊₁ oo . . . ^oo V _N ^(d−τ)

I _r ^(s) I _r+1 ^(s) I _N ^(s) to a resolution

X r X _r+1 X _N

V _r ^{(d) oo} V _r+1 ^(d) oo . . . ^oo V _N ^(d) Since f _i = x _i and

{x ∈ V ^(d) | ν x (x _i ) ≥ 1} = {x ₁ = 0, . . . , x τ = 0}

is naturally identified with V ^{(d−τ )} .

Strategy in positive characteristic

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

S TRATEGY IN POSITIVE CHARACTERISTIC

β : V ^(d) −→ V ^(d−1) −→ . . . −→ V ^(d−τ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f _τ ^(p

⁾ (z _τ ) = z _τ ^p

+ a ^(τ) ₁ z _τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

Strategy in positive characteristic

T HEOREM (S TAGE A): R EDUCTION TO THE MONOMIAL CASE .

X X ₁ X r

V ^(d) ←− V ₁ ^(d) ←− . . . ←− V _r ^(d)

is defined so that, setting

V ^{(d−τ )} ←− V ₁ ^(d−τ) ←− . . . ←− V _r ^(d−τ) and I ^(s) as above, then

I _r ^(s) = I(H ₁ ^(d−τ) ) ^α

· I(H ₂ ^(d−τ) ) ^α

. . . I(H _r ^(d−τ) ) ^α

⊂ O

V

.

T HEOREM (S TAGE B’): T HE MONOMIAL CASE .

Given I _r ^(s) = I(H ₁ ^(d−τ) ) ^α

· I(H ₂ ^(d−τ) ) ^α

. . . I(H _r ^(d−τ) ) ^α

⊂ O

V

, there exists (a τ -monomial):

I _r ^0(s) = I(H ₁ ^(d−τ) ) ^h

· I(H ₂ ^(d−τ) ) ^h

. . . I(H _r ^(d−τ) ) ^h

with exponents 0 ≤ h _i ≤ α _i , so that a combinatorial resolution:

V _p ^{(d−τ) oo} V _p+1 ^(d−τ) oo . . . ^oo V _N ^(d−τ) I _r ^0(s) I _r+1 ^0(s) I _N ^0(s)

can be lifted to a permissible sequence

X X ₁ X _N

V ^{(d) oo} V ₁ ^(d) oo . . . ^oo V _N ^(d) .

Strategy in positive characteristic

T HEOREM (S TAGE B’): T HE MONOMIAL CASE .

Given I _r ^(s) = I(H ₁ ^(d−τ) ) ^α

· I(H ₂ ^(d−τ) ) ^α

. . . I(H _r ^(d−τ) ) ^α

⊂ O

V

, there exists (a τ -monomial):

I _r ^0(s) = I(H ₁ ^(d−τ) ) ^h

· I(H ₂ ^(d−τ) ) ^h

. . . I(H _r ^(d−τ) ) ^h

with exponents 0 ≤ h _i ≤ α _i , so that a combinatorial resolution:

V _p ^{(d−τ) oo} V _p+1 ^(d−τ) oo . . . ^oo V _N ^(d−τ) I _r ^0(s) I _r+1 ^0(s) I _N ^0(s) can be lifted to a permissible sequence

X X ₁ X _N

V ^{(d) oo} V ₁ ^(d) oo . . . ^oo V _N ^(d) .

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Rees algebras and main invariants

B smooth over k ; V = Spec (B)

G = B[f ₁ W ⁿ

, f ₂ W ⁿ

, . . . , f _s W ⁿ

] = ⊕ _n≥0 I _n W ⁿ ⊂ B[W ]; I ₀ = O _V Sing(G) = {x ∈ V | ν _x (I _n ) ≥ n}

D EFINITION (H IRONAKA )

ord : Sing(G) → Q ord(x ) = min n ν x (I _k )

k o

Remark: ord(x ) = min n _ν

x

(f

)

n

, i = 1, . . . , s o

B smooth over k ; V = Spec (B)

G = B[f ₁ W ⁿ

, f ₂ W ⁿ

, . . . , f _s W ⁿ

] = ⊕ _n≥0 I _n W ⁿ ⊂ B[W ]; I ₀ = O _V Sing(G) = {x ∈ V | ν _x (I _n ) ≥ n}

D EFINITION (H IRONAKA )

ord : Sing(G) → Q ord(x ) = min n ν x (I _k )

k o

Remark: ord(x ) = min n _ν

x

(f

)

n

, i = 1, . . . , s o

Rees algebras and main invariants

B smooth over k ; V = Spec (B)

G = B[f ₁ W ⁿ

, f ₂ W ⁿ

, . . . , f _s W ⁿ

] = ⊕ _n≥0 I _n W ⁿ ⊂ B[W ]; I ₀ = O _V Sing(G) = {x ∈ V | ν _x (I _n ) ≥ n}

D EFINITION (H IRONAKA )

ord : Sing(G) → Q ord(x ) = min n ν x (I _k )

k o

Remark: ord(x ) = min n _ν

x

(f

)

n

, i = 1, . . . , s o

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Resolution of Rees algebras

Smooth center Y ⊂ Sing(G) = {x ∈ V | ν x (I n ) ≥ n}

V ←− ^π V ₁ H = π ⁻¹ (Y ) I n O

V

= I(H) ⁿ I _n ⁰ D EFINITION (T RANSFORMATION )

G ₁ = ⊕ _n≥0 I _n ⁰ W ⁿ is the transform of G = ⊕ _n≥0 I _n W ⁿ

(V , G) ^oo (V ₁ , G ₁ )

Y ⊂ Sing(G)

Smooth center Y ⊂ Sing(G) = {x ∈ V | ν x (I n ) ≥ n}

V ←− ^π V ₁ H = π ⁻¹ (Y ) I n O

V

= I(H) ⁿ I _n ⁰ D EFINITION (T RANSFORMATION )

G ₁ = ⊕ _n≥0 I _n ⁰ W ⁿ is the transform of G = ⊕ _n≥0 I _n W ⁿ

(V , G) ^oo (V ₁ , G ₁ )

Y ⊂ Sing(G)

Resolution of Rees algebras

R ESOLUTION

V smooth /k , G = ⊕I _k W ^k ⊂ O _V

Find a sequence of permissible transformations:

(V , G) ←− (V ₁ , G ₁ ) ←− . . . ←− (V _n , G _n )

with

Sing(G _n ) = ∅

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Rees algebras and integral closure

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

Rees algebras and integral closure

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

Rees algebras and integral closure

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

R EES A LGEBRAS AND INTEGRAL CLOSURE

G ₁ = ⊕ _n≥0 I _n W ⁿ , G ₂ = ⊕ _n≥0 J _n W ⁿ , ⊂ O _V

[W ] G ₁ ∼ G ₂ if same integral closure in O _V

[Z ] Then:

G ₁ ∼ O _V

[I _s W ^s ] for some s.

Sing(G ₁ ) = Sing(G ₂ ).

Hironaka’s functions coincide.

Equivalence is stable by transformations.

Multiplicity of Hypersurfaces

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

A smooth/k, V = Spec(A[Z ]) f = Z ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n ∈ A[Z ]

G = O _V [fW ⁿ ], Sing(G) = F _n = n-fold points of f.

Spec(A[Z ]) ^{β //} Spec(A)

Spec(A[Z ]/hf i)

OO 88 r r r r r r r r r r

β(F n )?

Multiplicity of Hypersurfaces

A smooth/k, V = Spec(A[Z ]) f = Z ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n ∈ A[Z ]

G = O _V [fW ⁿ ], Sing(G) = F _n = n-fold points of f.

Spec(A[Z ]) ^{β //} Spec(A)

Spec(A[Z ]/hf i)

OO 88 r r r r r r r r r r

β(F n )?

A smooth/k, V = Spec(A[Z ]) f = Z ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n ∈ A[Z ]

G = O _V [fW ⁿ ], Sing(G) = F _n = n-fold points of f.

Spec(A[Z ]) ^{β //} Spec(A)

Spec(A[Z ]/hf i)

OO 88 r r r r r r r r r r

β(F n )?

Multiplicity of Hypersurfaces

U NIVERSAL SETTING

(Z − Y ₁ )(Z − Y ₂ ) · · · (Z − Y _n ) ∈ k [Y ₁ , . . . , Y _n ][Z ] F _n (Z ) = Z ⁿ − s ₁ Z ⁿ⁻¹ + · · · + (−1) ⁿ s _n ∈ k [s ₁ , . . . , s _n ][Z ] k [s ₁ , . . . , s n ][Z ]/hF _n (Z )i ^// A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a n i

k [s ₁ , . . . , s _n ] ^//

OO

A

OO

s _i // (−1) ⁱ a _i .

U NIVERSAL SETTING

(Z − Y ₁ )(Z − Y ₂ ) · · · (Z − Y _n ) ∈ k [Y ₁ , . . . , Y _n ][Z ] F _n (Z ) = Z ⁿ − s ₁ Z ⁿ⁻¹ + · · · + (−1) ⁿ s _n ∈ k [s ₁ , . . . , s _n ][Z ] k [s ₁ , . . . , s n ][Z ]/hF _n (Z )i ^// A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a n i

k [s ₁ , . . . , s _n ] ^//

OO

A

OO

s _i // (−1) ⁱ a _i .

Multiplicity of Hypersurfaces

I NVARIANTS UNDER CHANGE OF VARIABLES

F _n (Z ) = (Z − Y ₁ )(Z − Y ₂ ) · · · (Z − Y _n ) ∈ k [Y ₁ , . . . , Y _n ][Z ] L = V (Y _i − Y _j , 1 ≤ i, j, ≤ n)

k[Y ₁ , . . . , Y _n ] ^L = k [Y _i − Y _j ; 1 ≤ i, j, ≤ n]

S _n acts linearly on k [Y ₁ , . . . , Y _n ] ^L ⊂ k [Y ₁ , . . . , Y _n ] (k [Y ₁ , . . . , Y n ] ^L ) ^S

⊂ (k [Y ₁ , . . . , Y n ]) ^S

(k[Y ₁ , . . . , Y _n ] ^L ) ^S

= k [H ₁ , . . . , H _r ]

H _j = H _j (Y ₁ , . . . , Y n ) homog of degree d _j

H _j = H _j (s ₁ , . . . , s _n ) w. homog. of degree d _j

I NVARIANTS UNDER CHANGE OF VARIABLES

F _n (Z ) = (Z − Y ₁ )(Z − Y ₂ ) · · · (Z − Y _n ) ∈ k [Y ₁ , . . . , Y _n ][Z ] L = V (Y _i − Y _j , 1 ≤ i, j, ≤ n)

k[Y ₁ , . . . , Y _n ] ^L = k [Y _i − Y _j ; 1 ≤ i, j, ≤ n]

S _n acts linearly on k [Y ₁ , . . . , Y _n ] ^L ⊂ k [Y ₁ , . . . , Y _n ] (k [Y ₁ , . . . , Y n ] ^L ) ^S

⊂ (k [Y ₁ , . . . , Y n ]) ^S

(k[Y ₁ , . . . , Y _n ] ^L ) ^S

= k [H ₁ , . . . , H _r ]

H _j = H _j (Y ₁ , . . . , Y n ) homog of degree d _j

H _j = H _j (s ₁ , . . . , s _n ) w. homog. of degree d _j

Multiplicity of Hypersurfaces

I NVARIANTS UNDER CHANGE OF VARIABLES

F _n (Z ) = (Z − Y ₁ )(Z − Y ₂ ) · · · (Z − Y _n ) ∈ k [Y ₁ , . . . , Y _n ][Z ] L = V (Y _i − Y _j , 1 ≤ i, j, ≤ n)

k[Y ₁ , . . . , Y _n ] ^L = k [Y _i − Y _j ; 1 ≤ i, j, ≤ n]

S _n acts linearly on k [Y ₁ , . . . , Y _n ] ^L ⊂ k [Y ₁ , . . . , Y _n ] (k [Y ₁ , . . . , Y n ] ^L ) ^S

⊂ (k [Y ₁ , . . . , Y n ]) ^S

(k[Y ₁ , . . . , Y _n ] ^L ) ^S

= k [H ₁ , . . . , H _r ]

H _j = H _j (Y ₁ , . . . , Y n ) homog of degree d _j

H _j = H _j (s ₁ , . . . , s _n ) w. homog. of degree d _j

k [s ₁ , . . . , s _n ][Z ]/hF _n (Z )i A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n i

k [s ₁ , . . . , s _n ]

OO

φ // (A, M)(loc., reg.)

OO

s _i ^// (−1) ⁱ a _i ∈ M _i . φ(H _j ) = H _j (a ₁ , . . . , a _n ) = h _j ∈ M ^d

hh _j i invariant by Z → uZ + a u ∈ U(A); a ∈ A.

p ∈ V (hh ₁ , . . . , h n

i) ⊂ Spec(A) iff B ⊗ k (p) is local.

∩{P ∈ Spec (A) : ν _P (h _i ) ≥ d _i } = β(F n ) if n 6= 0 in k .

Multiplicity of Hypersurfaces

k [s ₁ , . . . , s _n ][Z ]/hF _n (Z )i A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n i

k [s ₁ , . . . , s _n ]

OO

φ // (A, M)(loc., reg.)

OO

s _i ^// (−1) ⁱ a _i ∈ M _i . φ(H _j ) = H _j (a ₁ , . . . , a _n ) = h _j ∈ M ^d

hh _j i invariant by Z → uZ + a u ∈ U(A); a ∈ A.

p ∈ V (hh ₁ , . . . , h n

i) ⊂ Spec(A) iff B ⊗ k (p) is local.

∩{P ∈ Spec (A) : ν _P (h _i ) ≥ d _i } = β(F n ) if n 6= 0 in k .

k [s ₁ , . . . , s _n ][Z ]/hF _n (Z )i A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n i

k [s ₁ , . . . , s _n ]

OO

φ // (A, M)(loc., reg.)

OO

s _i ^// (−1) ⁱ a _i ∈ M _i . φ(H _j ) = H _j (a ₁ , . . . , a _n ) = h _j ∈ M ^d

hh _j i invariant by Z → uZ + a u ∈ U(A); a ∈ A.

p ∈ V (hh ₁ , . . . , h n

i) ⊂ Spec(A) iff B ⊗ k (p) is local.

∩{P ∈ Spec (A) : ν _P (h _i ) ≥ d _i } = β(F n ) if n 6= 0 in k .

Multiplicity of Hypersurfaces

k [s ₁ , . . . , s _n ][Z ]/hF _n (Z )i A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n i

k [s ₁ , . . . , s _n ]

OO

φ // (A, M)(loc., reg.)

OO

s _i ^// (−1) ⁱ a _i ∈ M _i . φ(H _j ) = H _j (a ₁ , . . . , a _n ) = h _j ∈ M ^d

hh _j i invariant by Z → uZ + a u ∈ U(A); a ∈ A.

p ∈ V (hh ₁ , . . . , h n

i) ⊂ Spec(A) iff B ⊗ k (p) is local.

∩{P ∈ Spec (A) : ν _P (h _i ) ≥ d _i } = β(F n ) if n 6= 0 in k .

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Elimination and differential operators

T AYLOR MORPHISM

Tay : A[Z ] → A[Z , T ] Z → Z + T Tay(f (Z )) = P

r ∈ N b r (X )T ^r ; ∆ ^r (f (X )) = b r (X )

p ∈ Spec(S[Z ]), if ν p (f (Z )) ≥ n, then ν p (∆ ^r (f (Z ))) ≥ n − r A[Z ][fW ⁿ ] ⊂ A[Z ][∆ ⁿ⁻¹ (f )W ¹ , . . . , ∆ ¹ (f )W ⁿ⁻¹ , fW ⁿ ]

Sing(A[Z ][fW ⁿ ]) = Sing(A[Z ][∆ ^k (f )W ^n−k , fW ⁿ ])

T AYLOR MORPHISM

Tay : A[Z ] → A[Z , T ] Z → Z + T Tay(f (Z )) = P

r ∈ N b r (X )T ^r ; ∆ ^r (f (X )) = b r (X )

p ∈ Spec(S[Z ]), if ν p (f (Z )) ≥ n, then ν p (∆ ^r (f (Z ))) ≥ n − r A[Z ][fW ⁿ ] ⊂ A[Z ][∆ ⁿ⁻¹ (f )W ¹ , . . . , ∆ ¹ (f )W ⁿ⁻¹ , fW ⁿ ]

Sing(A[Z ][fW ⁿ ]) = Sing(A[Z ][∆ ^k (f )W ^n−k , fW ⁿ ])

Elimination and differential operators

I NVARIANTS AND DIFFERENTIAL OPERATORS

F n (Z ) = Q

(Z − Y _i ) = Z ⁿ − s ₁ Z ⁿ⁻¹ + . . . s n ∈ k [s ₁ , . . . , s n ] k [Y ₁ , . . . , Y _n ] k [s ₁ , . . . , s _n ]

∪ −→ ^S

∪

k [Y _i − Y _j ] k[H ₁ , . . . , H _n

]

k [Z − Y ₁ , . . . , Z − Y _n ] k [∆ ^r F _n , F _n ]

∪ −→ ^S

∪

k [Y _i − Y _j ] k [H ₁ , . . . , H n

]

Each H _i is weighted hom. on the ∆ ^r F n (weight n − r ).

I NVARIANTS AND DIFFERENTIAL OPERATORS

F n (Z ) = Q

(Z − Y _i ) = Z ⁿ − s ₁ Z ⁿ⁻¹ + . . . s n ∈ k [s ₁ , . . . , s n ] k [Y ₁ , . . . , Y _n ] k [s ₁ , . . . , s _n ]

∪ −→ ^S

∪

k [Y _i − Y _j ] k[H ₁ , . . . , H _n

]

k [Z − Y ₁ , . . . , Z − Y _n ] k [∆ ^r F _n , F _n ]

∪ −→ ^S

∪

k [Y _i − Y _j ] k [H ₁ , . . . , H n

]

Each H _i is weighted hom. on the ∆ ^r F n (weight n − r ).

Elimination and differential operators

W EIGHTED P ULL - BACKS OF ALGEBRAS

Set k [Z − Y ₁ , . . . , Z − Y _n ] ^S

= k [∆ ^r F _n , F _n ]. The weighted pull-back k [∆ ^r F n W ^n−r , F n W ⁿ ] is

A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

The weighted-Pull-Back ofk [H ₁ W ^d

, . . . , H _s W ^d

] is A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[W ].

A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ] D EFINITION

A[h ₁ W ^d

, . . . , h _s W ^d

] elimination alg. of A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

W EIGHTED P ULL - BACKS OF ALGEBRAS

Set k [Z − Y ₁ , . . . , Z − Y _n ] ^S

= k [∆ ^r F _n , F _n ]. The weighted pull-back k [∆ ^r F n W ^n−r , F n W ⁿ ] is

A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

The weighted-Pull-Back ofk [H ₁ W ^d

, . . . , H _s W ^d

] is A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[W ].

A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ] D EFINITION

A[h ₁ W ^d

, . . . , h _s W ^d

] elimination alg. of A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

Elimination and differential operators

W EIGHTED P ULL - BACKS OF ALGEBRAS

Set k [Z − Y ₁ , . . . , Z − Y _n ] ^S

= k [∆ ^r F _n , F _n ]. The weighted pull-back k [∆ ^r F n W ^n−r , F n W ⁿ ] is

A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

The weighted-Pull-Back ofk [H ₁ W ^d

, . . . , H _s W ^d

] is A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[W ].

A[h ₁ W ^d

, . . . , h _s W ^d

] ⊂ A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ] D EFINITION

A[h ₁ W ^d

, . . . , h _s W ^d

] elimination alg. of A[Z ][f _n W ⁿ , ∆ ^r f _n W ^n−r ].

O UTLINE

1 I NTRODUCTION .

Multiplicity and the τ -invariant Local presentation

Strategy in characteristic zero Strategy in positive characteristic

2 R EES A LGEBRAS

Rees algebras and main invariants Resolution of Rees algebras Rees algebras and integral closure

3 E LIMINATION

Multiplicity of hypersurfaces Elimination and differentials

Absolute and relative differential structure

4 T HE MONOMIAL CASE .

Stage B’

Elimination: Absolute and relative differential structure

β d,d−τ : V ^(d) −→ V ^(d−τ) smooth τ ≤ d Diff _β ^r

d,d−τ

sheaf of relative differential operators.

D EFINITION

G = ⊕ _n≥0 I n W ⁿ ⊂ O _V

[W ] is a relative Diff-algebra if Diff _β ^r

d,d−τ

(I _n ) ⊂ I _n−r . Properties:

There is a smallest extension G ⊂ G _β

(G), where G _β

(G) is a relative (or absolute) Diff-algebra.

Sing(G) = Sing(G _β

(G)).

G _β

compatible with integral closure.

β d,d−τ : V ^(d) −→ V ^(d−τ) smooth τ ≤ d Diff _β ^r

d,d−τ

sheaf of relative differential operators.

D EFINITION

G = ⊕ _n≥0 I n W ⁿ ⊂ O _V

[W ] is a relative Diff-algebra if Diff _β ^r

d,d−τ

(I _n ) ⊂ I _n−r . Properties:

There is a smallest extension G ⊂ G _β

(G), where G _β

(G) is a relative (or absolute) Diff-algebra.

Sing(G) = Sing(G _β

(G)).

G _β

compatible with integral closure.

Elimination: Absolute and relative differential structure

β d,d−τ : V ^(d) −→ V ^(d−τ) smooth τ ≤ d Diff _β ^r

d,d−τ

sheaf of relative differential operators.

D EFINITION

G = ⊕ _n≥0 I n W ⁿ ⊂ O _V

[W ] is a relative Diff-algebra if Diff _β ^r

d,d−τ

(I _n ) ⊂ I _n−r . Properties:

There is a smallest extension G ⊂ G _β

(G), where G _β

(G) is a relative (or absolute) Diff-algebra.

Sing(G) = Sing(G _β

(G)).

G _β

compatible with integral closure.

β d,d−τ : V ^(d) −→ V ^(d−τ) smooth τ ≤ d Diff _β ^r

d,d−τ

sheaf of relative differential operators.

D EFINITION

G = ⊕ _n≥0 I n W ⁿ ⊂ O _V

[W ] is a relative Diff-algebra if Diff _β ^r

d,d−τ

(I _n ) ⊂ I _n−r . Properties:

There is a smallest extension G ⊂ G _β

(G), where G _β

(G) is a relative (or absolute) Diff-algebra.

Sing(G) = Sing(G _β

(G)).

G _β

compatible with integral closure.

Elimination: Absolute and relative differential structure

R ELATIVE DIFFERENTIALS AND E LIMINATION

Given G and V ^(d) −→ ^β V ^(d−1) smooth and generic.

Assume,

G = O _V

[f n W ⁿ , ∆ ^α (f n )W ^n−α ] 1≤α≤n

then G is a relative Diff-algebra and there is an elimination algebra,

R _G,β ⊂ O _V

[W ].

V smooth | _k G = ⊕I _k W ^k ⊂ O _V [W ] ord : Sing(G) → Q Assume: ord(x ) = 1 for any x ∈ Sing(G).

At each x ∈ Sing(G) : C G,x ⊂ T V ,x

Linear subspace L G,x ⊂ C G,x

D EFINITION (H IRONAKA ) τ (x )= codim L G,x in T V,x . D EFINITION

G is of codimensional type ≥ e if τ (x ) ≥ e for all x ∈ Sing(G)

Elimination: Absolute and relative differential structure

V smooth | _k G = ⊕I _k W ^k ⊂ O _V [W ] ord : Sing(G) → Q Assume: ord(x ) = 1 for any x ∈ Sing(G).

At each x ∈ Sing(G) : C G,x ⊂ T V ,x

Linear subspace L G,x ⊂ C G,x

D EFINITION (H IRONAKA ) τ (x )= codim L G,x in T V,x . D EFINITION

G is of codimensional type ≥ e if τ (x ) ≥ e for all x ∈ Sing(G)

V smooth | _k G = ⊕I _k W ^k ⊂ O _V [W ] ord : Sing(G) → Q Assume: ord(x ) = 1 for any x ∈ Sing(G).

At each x ∈ Sing(G) : C G,x ⊂ T V ,x

Linear subspace L G,x ⊂ C G,x

D EFINITION (H IRONAKA ) τ (x )= codim L G,x in T V,x . D EFINITION

G is of codimensional type ≥ e if τ (x ) ≥ e for all x ∈ Sing(G)

Elimination: Absolute and relative differential structure

V ^(d) smooth | _k G = ⊕I _k W ^k ⊂ O _V

Assume ord(x) = 1, and G is of codimensional type ≥ τ . T HEOREM

If G is rel. Diff-alg. for β d,d−τ : V ^(d) → V ^(d−τ) generic, then:

(Local Presentation)

G ∼ O _V

[f n

W ⁿ

, ∆ ^α

f n

W ⁿ

^−|α

^| ] β _d,d−e ^∗ (G ^(d−τ) ).

L OCAL PRESENTATION

(V ^(d) , x) −→ ^β (V ^(d−τ) , x _τ ) a composition of smooth morphisms (V ^(d) , x ) −→ (V ^(d−1) , x ₁ ) −→ . . . −→ (V ^(d−τ) , x _τ ) A local presentation of X (at x ) is defined by

(1) Positive integers 0 ≤ e ₁ ≤ e ₂ ≤ · · · ≤ e _τ . (2) Monic polynomials,

f ₁ ^(p

⁾ (z ₁ ) = z ₁ ^p

+ a ⁽¹⁾ ₁ z ₁ ^p

⁻¹ + · · · + a ⁽¹⁾ _p

∈ O _V

[z ₁ ] .. .

f τ ^(p

⁾ (z _τ ) = z τ ^p

+ a ^(τ) ₁ z τ ^p

⁻¹ + · · · + a ^(τ) _p

∈ O _V

[z _τ ].

(3) I ^(s) : an ideal in O _V

and a positive integer s.

F _b =

τ

\

i=1

{x ∈ V ^(d) | ν x f _i ^(p

⁾

≥ p ^e

}∩{x ∈ V ^(d) | ν x (β ^∗ (I ^(s) )) ≥ s}

Elimination: Absolute and relative differential structure

V ^(d) smooth | _k G = ⊕I _k W ^k ⊂ O _V

Assume ord(x) = 1, and G is of codimensional type ≥ e.

T HEOREM

If G is rel. Diff-alg. for β d,d−τ : V ^(d) → V ^(d−τ) generic, then:

(Local Presentation)

G ∼ O _V

[f n

W ⁿ

, ∆ ^α

f n

W ⁿ

^−|α

^| ] β _d,d−τ ^∗ (G ^(d−τ) ) G ^(d−τ) ⊂ O _V

[W ] (Elimination algebra).

Sing(G) = β d,d−τ (Sing(G)) ⊂ Sing(G ^(d−τ) )

The natural restriction ord : Sing(G ^(d−τ) ) → Q to Sing(G) is

independent of β d,d−τ .

V ^(d) smooth | _k G = ⊕I _k W ^k ⊂ O _V

Assume ord(x) = 1, and G is of codimensional type ≥ e.

T HEOREM

If G is rel. Diff-alg. for β d,d−τ : V ^(d) → V ^(d−τ) generic, then:

(Local Presentation)

G ∼ O _V

[f n

W ⁿ

, ∆ ^α

f n

W ⁿ

^−|α

^| ] β _d,d−τ ^∗ (G ^(d−τ) ) G ^(d−τ) ⊂ O _V

[W ] (Elimination algebra).

Sing(G) = β d,d−τ (Sing(G)) ⊂ Sing(G ^(d−τ) )

The natural restriction ord : Sing(G ^(d−τ) ) → Q to Sing(G) is

independent of β d,d−τ .

Elimination: Absolute and relative differential structure

T HEOREM

If G is absolute diff. algebra: Sing(G) = Sing(G ^(d−τ) ) (Stability) If Y ⊂ Sing(G) smooth, Y ∼ = π(Y ) and

(V ^(d) , G)

β

(V ₁ ^(d) , G ₁ )

oo

β

(V ^(d−τ) , G ^{(d−τ )} ) ^oo (V ₁ ^{(d−τ )} , G ₁ ^(d−τ) )

And G ₁ is relative differential.

T HEOREM

If G is absolute diff. algebra: Sing(G) = Sing(G ^(d−τ) ) (Stability) If Y ⊂ Sing(G) smooth, Y ∼ = π(Y ) and

(V ^(d) , G)

β

(V ₁ ^(d) , G ₁ )

oo

β

(V ^(d−τ) , G ^{(d−τ )} ) ^oo (V ₁ ^{(d−τ )} , G ₁ ^(d−τ) )

And G ₁ is relative differential.