Rees Algebras, Elimination, and singularities ofVarieties over Perfect Fields

2012 CMI Summer School “The Resolution of Singular Algebraic Varieties”Obergurgl, Tyrolean Alps, Austria

June, 2012

The Problem

Given an algebraic variety X defined over a field k ,

construct a resolution of singularities

as a composition of blow-ups at regular centers,

X←−X1←− . . .←−Xn.

Here:

I Xn is regular;

I X ← Xn is birational and proper.

To address the problem we need to:

1. Identify the worst singularities of X .

2. Give a criterion to select a regular center to blow-up.

3. Compare the worst singularities of X before and after blow-up.

4. Check that the singularities after each blow-up have improved.

1. Identify the worst singularities of X .

I Use the Hilbert-Samuel function, HS(X):

X is regular ⇐⇒ HS(X) is constant on X .

I Use the Multiplicity, Mult(X),

X is regular ⇐⇒Mult(X) = 1 on X .

Both HS(X), Mult(X):

• Are upper semi-continuous functions;

• Stratify X in locally closed strata.

Stratum corresponding to the maximum value Max HS(X),Max Mult(X):

Max HS(X); Max Mult(X).

Notation: Use F(X) to refer to either HS(X) or Mult(X); Max F(X) forthe maximum value, and set

Max F(X) = Max HS(X); or Max Mult(X).

Both HS(X), Mult(X):

• Are upper semi-continuous functions;

• Stratify X in locally closed strata.

Stratum corresponding to the maximum value Max HS(X),Max Mult(X):

Max HS(X); Max Mult(X).

Notation: Use F(X) to refer to either HS(X) or Mult(X); Max F(X) forthe maximum value, and set

Max F(X) = Max HS(X); or Max Mult(X).

2. Give a criterion to select a regular center to blow-up.

In general, Max F(X) will not be smooth.

Problem ]1: Find a refinement of the function F(X).

This will be done using:

I Local presentations;

I Defining new invariants via elimination.

3. Compare the worst singularities of X before and after blow-up.

Consider the blow-up at a regular center Y ⊂Max F(X):

X ← X1

Problem ]2: Compare Max F(X) with Max F(X1).

The comparison will be possible using local presentations.

FACT: Max F(X)≥Max F(X1)

4. Check that the singularities after each blow-up have improved.

Problem ]3:

What if F(X) = F(X1)?

How do we know there is an improvement?

The improvement can be detected via other invariants that can bederived from the local presentations using techniques of elimination.

Conclusion

When trying to find a resolution of X we have to face some problems,that can be approached via:

Local presentation in regular schemes.

What are “Local presentations”. . . ?

Hironaka’s approach: Local presentations for the Hilbert-Samuelfunction

I There is an embedding of X in a smooth V ;

Locally at ξ ∈Max HS(X):

I There are functions f1, . . . , fs on V and positive integersm1, . . . ,ms, naturally attached to the Max HS(X) so that

Max HS(X) =\

1≤j≤s

{ξ ∈ V : νξ(fj)≥mj} ⊂ V . (1)

Moreover, given any sequence of blow-ups

V = V0 ← V1 ← . . . ← Vn

∪ ∪ . . . ∪X = X0 ← X1 ← . . . ← Xn

(2)

with regular centers Yi ⊂Max HS(Xi), one has that:

Max HS(Xi) = ∩{η ∈ Vi : νη(f (i)j )≥mj} ⊂ Vi (3)

where f (i)j denotes a weighted transform of f (i−1)

j for i = 1, . . . ,n.

Local presentations for the multiplicity

Work locally at ξ ∈Max Mult(X):

I There is a finite (local) morphism onto a smooth V , β : X → V ;with dim X = dim V .

I There are functions f1, . . . , fs on V and positive integersm1, . . . ,ms, naturally attached to the Max Mult(X) so that:

β(Max Mult(X)) = ∩{ξ ∈ V : νξ(fj)≥mj} ⊂ V .

Moreover, given any sequence of blow-ups with regular centersYi ⊂Max Mult(Xi), there is a commutative diagram of blow-ups andfinite projections:

X = X0

��

oo · · · Xnoo

��V = V0 oo · · · Vnoo

(4)

and for i = 0, . . .n one has that:

βi(Max Mult(Xi)) = ∩{η ∈ Vi : νη(f (i)j )≥mj} ⊂ Vi

where f (i)j denotes a weighted transform of f (i−1)

j for i = 1, . . . ,n.

Conclusion

We consider two strategies to describe Max F(X):

I For the Hilbert-Samuel function: use local embeddings in smoothschemes;

I For the Multiplicity: use local finite projections onto smoothschemes.

In both cases, Max F(X) is described via weighted equations.

To handle this of information we will use Rees algebras on smoothschemes.

The role of Rees algebras

I Associate a Rees algebra to a given local presentation.

I A resolution of the Rees algebra will lead to a lowering ofMax F(X).

I Max F(X) can be stratified in smooth strata via invariants definedusing Rees algebras using elimination.

I When the characteristic is zero, Rees algebras can be resolvedusing induction in the so called codimensional type, τ.

Moreover we will see that:

I The string of invariants for resolution, defined in terms of a Reesalgebra, will not depend on the local embedding, or the choice ofthe local finite projection.

I The codimensional type of a Rees algebra will be our tool toknow how many variables can be eliminated; this will play a rolein inductive arguments.

I The Canonicity Principle for Rees algebras will be our tool toglobalize local invariants.

Problem

Let dim X = d .

Consider upper-semi-continuous functions

F(X) : X → (Λ,≥)

such that if Y ⊂MaxF(X) is smooth, then after blowing-up at Y ,

V ← V1

one has thatMax F(X)≥Max F(X1).

Representability in dimension d

We will say that F(X) is representable in dimension d if, locally, ateach ξ ∈MaxF(X), there is a morphism:

β : X → V

with V smooth and d-dimensional such that:

I Max F(X) is homeomorphic to β(Max F(X));

I There is a Rees algebra G on V such that

SingG = β(Max F(X)).

Moreover, any sequence of transformations

X ← X1← . . .← Xn

at smooth centers Yi ⊂Max F(Xi), induces commutative diagrams ofmorphisms and blow-ups, and transforms of Rees algebras,

X = X0

��

oo · · · Xnoo

��V = V0 oo · · · Vnoo

G = G0 · · · Gn

such thatβi(Max F(Xi)) = SingGi

for i = 0, . . . ,n.

Examples

I Multiplicity (Villamayor’s course)

I Hilbert-Samuel (this can be shown);

I Others. . .

What if there are two different representations?