Birational models with normal crossingssingularities

Edward Bierstone

The Fields Institute

Universite de Nice Sophia AntipolisMay 17, 2011

Collaborators

Sergio Da Silva Pierre Lairez Pierre Milman Franklin VeraPacheco

General problem

Objects of study X ; e.g.,(1) (reduced) algebraic varieties X = X (dim perhaps given)(2) pairs X = (X ,D), where D is a Weil divisor on X

Morphisms σ : X ′ → X :induced by birational morphisms σ : X ′ → X

Can we find the smallest class of singularities S such that:

(1) S includes all normal crossings singularities

(2) Given X , there is a proper birational morphism σ : X ′ → Xsuch that

(a) X ′ has only singularities in S,(b) σ is an isomorphism over the nc locus of X ?

General problem

Objects of study X ; e.g.,(1) (reduced) algebraic varieties X = X (dim perhaps given)(2) pairs X = (X ,D), where D is a Weil divisor on X

Morphisms σ : X ′ → X :induced by birational morphisms σ : X ′ → X

Can we find the smallest class of singularities S such that:

(1) S includes all normal crossings singularities

(2) Given X , there is a proper birational morphism σ : X ′ → Xsuch that

(a) X ′ has only singularities in S,(b) σ is an isomorphism over the S-locus of X ?

Why ask these questions?

Resolution of singularities: In characteristic zero, every varietyis birationally equivalent to a smooth variety.

But birational models with mild singularities have to be admittedin natural situations.

Example. Family of projective curves

z3 + y3 + x3 − 3λxyz = 0,

smooth if λ3 6= 1. When λ = 1:

(z + y + x)(z + εy + ε2x)(z + ε2y + εx) = 0, ε3 = 1

We cannot simultaneously resolve the singularities of a familyof curves without allowing special fibres that have normalcrossings singularities.

Example. Whitney umbrella or pinch point pp

z2 + xy2 = 0

!2

!1

0

1

2

!2

!1

0

1

2

!2!1

0

1

2

pp at the origin, nc2 along non-zero x-axis

There is no proper birational morphism that eliminates ppwithout modifying nc points

Normal crossings

In local coordinates (x1, . . . , xn),

xα11 · · · x

αnn = 0

Local coordinates ?regular parameters simple normal crossings sncor local analytic coordinates (after finite field extension)

normal crossings nc

Examples

y2 = x2 + x3 nc, not snc

y2 + x2 = 0 nc, snc iff√−1 ∈ K

Normal crossings in pairs

(X ,D) is simple normal crossings sncif X is smooth and D is an snc divisor

(X ,D) is semi-simple normal crossings ssnc

if, locally, in regular coordinates:

X : x1 · · · xp = 0D is the restriction to X of

yα11 · · · y

αqq = 0

We will see that S = {normal crossings}in the case of either snc or ssnc

The philosophy

The desingularization invariant invtogether with natural geometric informationcan be used to compute local normal forms of singularities

Resolution of singularities

X = X0σ1←− X1 ←− · · ·

σt←− Xt = X ′

The centre of each blowing-up is the maximum locus of anupper-semicontinuous invariant inv defined recursivelyover a sequence of admissible blowings-up.

Examples

nc2 z2 + y2 = 0 inv(nc2) = (2,0,1,0,∞)

pp z2 + xy2 = 0 inv(pp) = (2,0,3/2,0,1,0,∞)

Lemma

nc2 ⇐⇒ inv = inv(nc2)

pp ⇐⇒ inv = inv(pp) and codim Sing X = 2

This is in year zero

The invariant depends on the history of blowings-up

E.g., in 3 variables, (inv = inv(nc2)) is a smooth curve.

It is generically nc2. But, at a special point:

z2 + wαy2 = 0, (w = 0) exceptional divisor

Cleaning: Blow up (z = w = 0).

In the coordinate chart (w , y ,wz), we get

w2(z2 + wα−2y2) = 0

Eventually,

w∗(z2 + y2) = 0, α even

w∗(z2 + wy2) = 0, α odd

Theorem. Normal forms of singularities in S

In two variables:

xy = 0 nc2

In three variables:

xy = 0xyz = 0 nc3

z2 + xy2 = 0 pp = cp2

nc singularities are singularities of hypersurfaces

Desingularization of pairs (X ,D) preserving sncFirst consider X smooth.

(1) At a given point in year zero,

D is snc(q) ⇐⇒ q componentsand inv = (q,0,1,0, · · · ,1,0,∞)

where there are q − 1 pairs (1,0).

(2) In general, RHS =⇒ local normal form

x1 (ξ2 + wα2x2) · · · (ξq + wαq xq) = 0

where ξj ∈ (x1, . . . , xj−1), wαj is an exceptional monomial,and αj ≤ αj+1.

Log resolution preserving snc

Transform (X ′,D′) of (X ,D)

by a birational morphism σ : X ′ → X :

D′ := birational transform of D + Ex(σ)

Theorem

Suppose that X is smooth.There is a morphism σ : X ′ → X given by a sequence ofadmissible blowings-up, such that

(a) (X ′,D′) is snc;(b) σ is an isomorphism over the snc-locus of (X ,D).

Moreover, we can avoid blowing up snc at every step.

Semi-log resolution preserving ssnc

Transform (X ′,D′) of (X ,D)

by a birational morphism σ : X ′ → X :

D′ := birational transform of D + Ex(σ)

Theorem

In general:There is a morphism σ : X ′ → X given by a sequence ofadmissible blowings-up, such that

(a) (X ′,D′) is ssnc;(b) σ is an isomorphism over the ssnc-locus of (X ,D).

Moreover, we can avoid blowing up ssnc at every step.

Semi-simple normal crossings

In higher codimension,inv begins with the Hilbert-Samuel function

Let Hp,q denote the Hilbert-Samuel function of

x1 · · · xp = 0, y1 · · · yq = 0

Exercise. In year zero, if (X ,D) is ssnc, then

invSupp D = (Hp,q,0,1,0, . . . ,1,0,∞) ,

where there are pairs (1,0).

Examples

X = (x1x2 = 0) ⊂ A4(x1,x2,y ,z)

Set Xi := (xi = 0), Di := D ∩ Xi .

If (X ,D) is ssnc, then

X1 ∩ D2 = X2 ∩ D1, codimX Supp D1 ∩ Supp D2 = 2

(1) D = (x1 = y = 0) + (x2 = z + x1 + yz = 0). Then

D1 ∩ D2 = (x1 = x2 = y = z = 0), codimX D1 ∩ D2 = 3.

Ideal of Supp D is (x1, y) · (x2, z + x1 + yz);order = 2, so HSupp D 6= H2,1.

In general, HSupp D = H2,1 =⇒ codimX Supp D1 ∩ Supp D2 = 2

Examples

(2) D = (x1 = y = 0) + (x2 = x1 + yz = 0). Then

X1 ∩ D2 = (x1 = x2 = yz = 0)6= (x1 = x2 = y = 0) = X2 ∩ D1.

codimX Supp D1 ∩ Supp D2 = 2and HSupp D = H2,1.

Note that

X1 ∩ D2 = X2 ∩ D1 + extra component (x1 = x2 = z = 0).

This is the worst that can happen when HSupp D = H2,1.

Minimal singularities of algebraic varieties

Theorem. The class S in four variables

xy = 0xyz = 0

xyzw = 0 nc4

z2 + xy2 = 0 (or z2 + (y + 2x)(y − x)2 = 0)

z2 + (y + 2x2)(y − x2)2 = 0 degenerate pinch point dpp

x(z2 + wy2) = 0 prod

z3 + wy3 + w2x3 − 3wxyz = 0 cyclic point singularity cp3

Limits of nc3 singularities in four variables

Consider

f (w , x , y , z) = z3 + a(w , x , y)z2 + b(w , x , y)z + c(w , x , y),

nc3 on the non-negative w-axis.

Suppose inv(0) = inv(nc3) = (3,0,1,0,1,0,∞).

At the origin:

f has 3 factors f has 2 factors f irreducible

f (w2, x , y , z) splits f (w3, x , y , z) splits

After cleaning:

f = xyz nc3 f = x(z2 + wy2) f = cp3

Local normal forms

Equivalent conditions

(1) f has two analytic factors;

(2) f (w2, x , y , z) splits (though f does not);

(3) after an etale coordinate change, either

f = (z + wαx)(

z2 + w2α+1(

xξ + wβy)2),

where ξ = ξ(w , x , y), or

f =(

z + wα(

yη + wβx))(

z2 + w2α+1y2),

where η = η(w , x , y).

Equivalent conditions

(1) f is analytically irreducible;

(2) f (w3, x , y , z) splits (though f does not);

(3) after an etale coordinate change,

f = z3 − 3wβy (yη + wγx) z +wαy3 +w3β−α (yη + wγx)3 ,

where η = η(w , x , y) and α is not divisible by 3.

Abhyankar-Jung phenomena

Limits of nc(k) singularities in n variables

Consider

f (−→w ,−→x ,−→y , z) = zk + a1(w , x , y)zk−1 + · · ·+ ak (w , x , y),

and (inv = inv(nc(k))). Assume f is generically nc(k) on

(inv = inv(nc(k))) = (y = z = 0) ,

and w = (w1, . . . ,wp) exceptional.

Question. Does f (w i11 , . . . ,w

ipp , x , y , z) split, for suitable im

(perhaps after exceptional monomial blowings-up)?

Questions

Abhyankar-Jung after normalizationConsider

f (−→y , z) = zk +k∑

j=1

aj(y)zk−j

Normal crossings locus ⊂ (z = 0),or splitting locus ⊂ (z = 0):

points over which f is normal crossings (or f splits)

Question. Suppose the non-nc locus (or the non-splittinglocus) is nc; say (wα = 0), where −→y = (

−→w ,−→x ).

Does f (w i11 , . . . ,w

ipp , x , z) split, for suitable im ?