III S INGULARITIESINPOSITIVECHARACTERISTIC

S INGULARITIES IN POSITIVE CHARACTERISTIC

III

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

Universidad Autónoma de Madrid

RIMS Kyoto, December 2008

O UTLINE

1 T RANSVERSALITY AND ELIMINATION ALGEBRA . Multiplicity and transversality

Elimination algebra

2 O VERVIEW

Multiplicity and transversality

O UTLINE

1 T RANSVERSALITY AND ELIMINATION ALGEBRA . Multiplicity and transversality

Elimination algebra

2 O VERVIEW

Multiplicity and transversality

M ULTIPLICITY

A smooth k -algebra.

f (Z ) = Z ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n ∈ A[Z ], F n = {x ∈ Spec(A[Z ]) | ν x (f ) ≥ n}. Set B = A[Z ]/f (Z ).

F n ⊂ Spec(B)

β

k

β(F n ) ⊂ Spec(A) Zariski’s multiplicity formula.

[B : A]e(q) = X

i≥1

[k (P _i ) : k (q)]e _B

Pi

(qB _P

).

Multiplicity and transversality

M ULTIPLICITY

A smooth k -algebra.

f (Z ) = Z ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a _n ∈ A[Z ], F n = {x ∈ Spec(A[Z ]) | ν x (f ) ≥ n}. Set B = A[Z ]/f (Z ).

F n ⊂ Spec(B)

β

k

β(F n ) ⊂ Spec(A) Zariski’s multiplicity formula.

[B : A]e(q) = X

i≥1

[k (P _i ) : k (q)]e _B

Pi

(qB _P

).

Multiplicity and transversality

T RANSVERSALITY

V ^(d) = Spec(A[Z ]) and V ^(d−1) = Spec(A) V ^(d) −→ ^β V ^(d−1)

X = Spec(A[Z ]/hZ ⁿ + a ₁ Z ⁿ⁻¹ + · · · + a n i) x ∈ F n ⊂ X

 // V ^(d)

}}{{{ {{{ {{

V ^(d−1)

L _X,x ⊂ C _{X ,x} ⊂ T _V

_,x

D EFINITION

β is transversal to X in x if the tangent line ` to β ⁻¹ (β(x)) is not

included in L _X,x .

Multiplicity and transversality

If C is smooth and C ⊂ F n ⊂ X , then L _C,x ⊂ L _X,x and

C ∼ = β(C)

X X ₁

V ^(d)

β

V ₁ ^(d)

π

oo

β

V ^(d−1) V ₁ ^(d−1)

π

oo

Multiplicity and transversality

V ^(d) −→ ^β V ^(d−1)

G ⊂ O _V

[W ], x ∈ Sing(G) ⊂ V ^(d) , τ G,x ≥ 1.

L _G,x ⊂ C _G,x ⊂ T _V

_,x

D EFINITION

β is transversal to G in x if the tangent line ` to β ⁻¹ (β(x)) is not included in L _G,x .

G G ₁

C ⊂ Sing(G) ⊂ V ^(d)

β

V ₁ ^(d)

π

oo

β

k

β(C) ⊂ V ^(d−1) V ₁ ^(d−1)

π

oo

Multiplicity and transversality

V ^(d) −→ ^β V ^(d−1)

G ⊂ O _V

[W ], x ∈ Sing(G) ⊂ V ^(d) , τ G,x ≥ 1.

L _G,x ⊂ C _G,x ⊂ T _V

_,x

D EFINITION

β is transversal to G in x if the tangent line ` to β ⁻¹ (β(x)) is not included in L _G,x .

G G ₁

C ⊂ Sing(G) ⊂ V ^(d)

β

V ₁ ^(d)

π

oo

β

k

β(C) ⊂ V ^(d−1) V ₁ ^(d−1)

π

oo

Elimination algebra

O UTLINE

1 T RANSVERSALITY AND ELIMINATION ALGEBRA . Multiplicity and transversality

Elimination algebra

2 O VERVIEW

Elimination algebra

V ^(d) −→ ^β V ^(d−1) i) G ⊂ O _V

[W ] so that τ G ≥ 1

ii) G is a β-relative differential algebra.

iii) β transversal at x .

We now define a Rees algebra, say R _G,β ⊂ O _V

[W ], such that

G Sing(G) ⊂ V ^(d)

β

β(Sing(G)) ⊂ Sing(R _G,β ) ⊂ V ^(d−1)

R _G,β

Elimination algebra

E LIMINATION AND PERMISSIBLE TRANSFORMATIONS

C ⊂ Sing(G)

G G ₁

C ⊂ Sing(G) ⊂ V ^(d)

β

V ₁ ^(d)

π

oo

β

β(C) ⊂ Sing(R _G,β ) ⊂ V ^(d−1) V ₁ ^(d−1)

π

oo

R G,β (R _G,β ) ₁

Elimination algebra

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

E LIMINATION ALGEBRA

k [Y _i − Y _j ; 1 ≤ i, j, ≤ n] ⊂ k [Z − Y ₁ , . . . , Z − Y _n ]

k[H ₁ W ^d

, . . . , H r W ^d

] ⊂ k [F n (Z )W ⁿ , ∆ ^α (F n (Z ))W ^n−α ] 1≤α≤n−1

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 .

G = A[Z ][f _n (Z )W ⁿ , ∆ ^α (f _n (Z ))W ^n−α ] _{1≤α≤n−1} ⊂ A[Z ][W ]

∪ ∪

R _G,β ⊂ A[W ]

T HEOREM (S TAGE A)

If τ G ≥ e there is a well defined sequence of permissible transformations:

(V ^(d) , G)

β

. . .

oo (V ^(d) _r , G _r ) ^oo

β

(V ^(d−e) , G ^(d−e) ) oo . . . ^oo (V _r ^(d−e) , G _r ^(d−e) ) such that Sing(G _r ) = ∅ or G _r ^(d−e) is monomial:

G _r ^(d−e) ∼ O

V

[(I(H ₁ ) ^α

· · · I(H _r ) ^α

)W ^s ]

β : V ^(d) −→ V ^(d−e) smooth locally at x, G, τ G,x ≥ e.

Assume β is a composition of smooth morphisms V ^(d) −→ V ^(d−1) −→ . . . −→ V ^(d−e)

L EMMA

Fix G ⊂ O _V [W ]. If τ G ≥ 1 and codimension of Sing(G) is 1 in V , then there exists Z (⊂ V ) smooth hypersurface so that

G ∼ O _V [I(Z )W ].

T HEOREM (S TAGE A)

Assume τ G ≥ 1. There is a sequence of permissible transformations

G G ₁ G _r

V ^(d)

β

V ₁ ^(d)

π

oo

β

. . .

oo V ^(d) _r

π

oo

β

V ^(d−1) V ₁ ^(d−1)

π

1)

oo . . . oo V ^(d−1) _r

π

oo

R _G,β (R _G,β ) ₁ (R _G,β ) r

Sing(G _r ) = ∅ or (R _G,β ) r = I(H ₁ ) ^α

. . . I(H _r ) ^α

W ^s

G _r G _r+1 G _R

V _r ^{(d) oo} V _r ^(d) ₊₁ oo . . . ^oo V _R ^(d)

R EFERENCES

A. Benito and O. Villamayor, ‘Monoidal transformations of singularities in positive characteristic’

http://arXiv.org/abs/0811.4148 26 November 2008.

A. Bravo and O. Villamayor, ‘Hypersurface singularities in positive characteristic and stratification of singular locus’.

http://arXiv.org/abs/0807.4308 27 July 2008.

O. Villamayor, ‘Hypersurface singularities in positive characteristic.’ Advances in Mathematics 213 (2007) 687-733.

O. Villamayor U. ‘Elimination with apllications to

singularities in positive characteristic’. Publications of

RIMS, Kyoto University. Vol 44, No. 2, 2008.