Singularities of Varieties

Singularities of Varieties
Let K be R or C. Let n > 0 be an integer. Let U be an open subset of
K n including o = (0, ...,0). A function f : U → K is called analytic at o
if it is infinitely often differentiable and there is a neighborhood of o where
it coincides with its Taylor series.
An analytic variety V ⊂ U is defined as the common zero of a set of analytic
functions. A germ of a variety is a class of pairs (U,V ), U open in K n
including o and V a subvariety, modulo the equivalence relation generated
by restriction to a smaller open subset.
A singularity of a variety is defined as the germ of a variety.
VO AAG 1

The Vanishing Ideal
Let R := K[[x 1 , ...,x n ]] be the ring of convergent power series. Let
V ⊆ U ⊆ K n be a singularity. Then Ideal(V ) is defined as the ideal of all
f ∈ R that vanish at all points in V . It does not depend on the choice of
the representative.
Any ideal of R is finitely generated. Vanishing ideals are radical. The
converse holds for the case K = C, but not for the case K = R.
VO AAG 2

Analytic Equivalence
An analytic map from a variety X to a variety Y is a function f : X → Y
whose coordinates are analytic functions. Two varieties are analytically
equivalent iff there are analytic maps f : X → Y and g : Y → X such that
fg = id Y and gf = id X .
Two singularities are analytically equivalent iff there exists an analytic
isomorphism between representatives taking o to o.
Two singularities V 1 and V 2 are analytically equivalent iff their quotient
algebras R 1 /Ideal(V 1 ) and R 2 /Ideal(V 2 ) are isomorphic as K-algebras.
VO AAG 3

Automorphisms of (K,0)
A convergent power series P ∈ K[[x]], P = P 0 + P 1 x + P 2 x 2 + P 3 x 3 + . ..
defines an automorphism of (K,0) iff P 0 = 0 and P 1 ≠ 0.
For any power series Q ∈ K[[x]] of positive order m, Q = Q m x m +
Q m+1 x m+1 +· · ·+, there is an automorphism P such that Q(P(x)) = ±x m .
In case K = C, we may omit the sign.
VO AAG 4

Smooth singularities
A variety of dimension n is called smooth or nonsingular at some point p iff
its germ at p is isomorphic to the germ of K n .
The Jacobian ideal is defined as the ideal generated by all (N − n)-minors
of the Jacobi matrix of a set of equations generating the vanishing ideal.
(N is the number of variable, n the dimension.)
Theorem. A variety is smooth at a point p iff p does not belong to the
zero set of the Jacobian ideal.
VO AAG 5

Reducible Singularities
An irreducible singularity is one which is not a union of two strictly smaller
singularities.
Example: The zero set of y 2 − x 2 − x 3 at the origin is the union of two
smooth analytic curve germs, namely the zero set of y + x + 1 2 x2 − 1 8 x3 +
1
16 x4 − 5
128 x5 . .. and the zero set of y − x − 1 2 x2 + 1 8 x3 − 1 16 x4 + 5
128 x5 ... .
VO AAG 6

Plane Curve Singularities
A plane curve singularity is given by a squarefree power series F(x,y)
vanishing at o. The order of F is invariant under analytic isomorphisms.
The singularity is smooth iff ord(F) = 1.
If a plane curve singularity is reducible, then its order is equal to the sum of
orders of the components.
VO AAG 7

Weierstraß Preparation
Theorem. Any plane curve singularity of order n is isomorphic to one
defined by an equation F ∈ K[[x]][y] (i.e. F is polynomial in y) such that
deg y (F) = n and coeff y n(F) = 1.
Sketch of Proof. First, we apply an isomorphism to transform the equation
F 0 of order n to another equation F 1 which is regular in y. This means
that if we write F 1 as power series in y with coefficients in K[[x]], then the
constant term of coeff y n(F 1 ) is not zero. This can be achieved by a linear
map (x, y) ↦→ (x + αy, y) for suitable α ∈ K.
Second, we write y n as F 1 Q + R with Q ∈ K[[x,y]] and R ∈ K[[x]][y],
deg y (R) < n, using power series division. Then y n − R and F 1 generate
the same ideal in K[[x,y]], hence y n − R is an equation as required.
VO AAG 8

Exercise
Prove that every double point is analytically equivalent to y 2 − x n for some
n ≥ 2.
VO AAG 9

Classification of Plane Double Points
How can we prove that y 2 − x n and y 2 − x m are not equivalent, for
2 ≤ m < n?
VO AAG 10

Generalizations
A hypersurface singularity in K d is given by a squarefree power series in
K[[x 1 ,...,x d ]].
A hypersurface singularity is smooth iff its order is 1.
The order of a reducible hypersurface singularity is equal to the sum of the
orders of the components.
Any hypesurface singularity of order n is isomorphic to one defined by
an equation F ∈ K[[x 1 , ...,x d−1 ]][x d ] such that that deg xd
(F) = n and
coeff xd n(F) = 1.
VO AAG 11

Generalizations
The Hessian of F is the symmetric d × d matrix of all second derivatives.
The rank of the Hessian and, in the real case, the absolute value of
the index (number of positive minus number of negative eigenvalues) are
analytic invariants.
Let F be the equation of a hypersurface singularity in K d such that the rank
of the Hessian is equal to r and the number of positive eigenvalues is s ≤ r/2
(without loss of generality, otherwise we multiply the equation by −1). Then
F is analytically equivalent to x 2 1 + · · · + x 2 s − x 2 s+1 − x 2 r + G(x r+1 , ...,x d )
for some G ∈ K[[x r+1 ,...,x d ]].
This decomposition is unique in the sense that r and s and the analytic
equivalence class of G as a singularity in K d−r are uniquely determined.
VO AAG 12

Deciding Equivalence
Given two squarefree power series F and F 1 , how can we decide whether
the singularities of F and F 1 are equivalent?
We will show that this can be decided by an algorithm who looks only at
finitely many coefficients of F and F 1 .
It follows that every plane curve singularity can be represented by a
polynomial.
VO AAG 13

Tougeron Theory
Let F, G be two power series of order at least 3. We say “G is close to F”
iff G − F is in J(F) 2 , the square of the Jacobian ideal.
Lemma. If G is close to F, then J(F) = J(G).
It follows that “close to” is an equivalence relation.
VO AAG 14

Tougerons Theorem
Theorem. If G is close to F, then G and F are analytically equivalent.
Proof. Given G close to F, we define an analytic isomorphism U :
(x, y) ↦→ (x + S(x,y), y + T(x,y)) such that F is close to F ◦ U and
ord(F ◦ U − G) > ord(F − G). Then we show that the composition of
these isomorphisms converges.
VO AAG 15

Finite Determinacy
Theorem. There exists an integer n, depending on F, such that J(F) 2
contains all power series of order greater than n.
Proof. Assume, indirectly, that ∂ x F and ∂ y F have a common irreducible
factor G. Then F is constant in the zero set of G, hence G divides F,
F = GH for some H, and G divides G x H, hence F is not squarefree.
This shows that the zero set of J(F) is just the point o. By Hilberts
Nullstellensatz, the radical of J(F) is equal to 〈x, y〉. Hence there exists
m such that x m ∈ J(F) and y m ∈ J(F). It follows that any monomial of
order at least n = 3m − 1 is in J(F) 2 .
VO AAG 16

Exercise
Determine N such that x 3 + y 3 is equivalent to x 3 + y 3 + G(x, y) for any
G of order greater than n.
VO AAG 17

Deformations
A deformation of a plane curve singularity F(x, y) is a germ of an analytic
function G(x, y, ǫ) such that G(x, y, 0) = F(x,y).
Theorem. The order of a singularity does not increase in a deformation.
Problem. Which isomorphism classes arise in a deformation of a fixed
singularity? Solve this problem for F(x,y) = y 2 − x n .
VO AAG 18

Group Actions
A Lie group is a nonsingular variety (manifold) such that the multiplication
and the inverse are analytic functions.
An (left) action of a Lie group G on a manifold M is an analytic function
G × M → M, (g,m) ↦→ g.m, such that (gh).m = g.(h.m).
The group action induces an equivalence relation on M. The equivalence
classes are called orbits.
VO AAG 19

Group Actions
Theorem. Let N ⊆ N be a submanifold. Let p ∈ N. If the tangents
space of N and the image of the tangent space of G under the differential
of the map G → M, g ↦→ g.p, generate the tangent space of M, then N
intersects every orbit in a neighborhood of p.
VO AAG 20

Application
Let F(x, y) be a polynomial, and let n be an integer such that F(x,y) +
G(x, y) ≡ F(x, y) for any G or order greater than n. We consider the group
G 1 × G 2 , where G 1 is the group of all analytic isomorphisms truncated at
order n, and G 2 is the group of all units in the power series ring, truncated
at order n. The set M is the set of all power series truncated at order n,
G 1 acts by substitution and G 2 by multiplication.
VO AAG 21

Examples
F(x, y) = x 3 + y 3 ,
M = {x 3 + y 3 + a ij x i y j | 1 ≤ i + j ≤ 5,a ij ∈ K},
TM = {a ij x i y j | 1 ≤ i + j ≤ 5},
G 1 = {(u ij x i y j ,v ij x i y j ) | 1 ≤ i + j ≤ 5,u 10 v 01 − u 01 v 10 ≠ 0} ,
TG 1 = {(u ij x i y j , v ij x i y j ) | 1 ≤ i + j ≤ 5} ,
G 2 = {w ij x i y j | 0 ≤ i+j ≤ 5, w 00 ≠ 0}, TG 2 = {w ij x i y j | 0 ≤ i+j ≤ 5},
(TG 1 × TG 2 ) → TM, (U,V,W) ↦→ 3x 2 U + 3y 2 V + (x 3 + y 3 )W
The image of the map is the ideal generated by δ x F,δ y F,F. Any vectorspace
in TM that contains representatives for all residue classes satisfies the
assumption of the theorem.
VO AAG 22

Problems
Which isomorphism classes arise in deformations of x 3 + y 3 ?
Same question for x 2 y + y 4 , x 3 + y 4 , x 3 + y 5 , x 3 + y 6 .
VO AAG 23

Higher Dimensional Analogues
Tougeron’s theorem is valid in any dimension.
What fails to hold is finite determinacy. There exists no N such that for all
F ∈ K[[x, y, z]] of order larger than N, x 2 +y 2 +F is equivalent to x 2 +y 2 .
Finite determinacy does hold for isolated singularities.
VO AAG 24

Singularities of Maps
Let M, N be real differential manifolds (maybe with boundary). The set
D(M,N) of differentiable functions admits several topologies. We can work
with any topology such that the Taylor expansion at any point is continuous.
Let us write D ∗ (M) for the set of all diffeomorphisms from M to itself.
Then D ∗ (M) × D ∗ (N) acts on D(M, N) by (u,v).f = vfu −1 .
VO AAG 25

Stable Maps
A map f : M → N is stable iff its orbit under the action above is open in
D(M,N). This means that any sufficiently close map is equivalent to it.
Example. M = N = [−1,1]. Characterize all stable maps.
In this example, the stable maps are dense, i.e. any map can be deformed
to a stable map.
VO AAG 26

Morse Lemma
Assume N = R. Then f is stable iff the Hessian is nondegenerate at all
critical points, and the function values of the critical points ar all different.
Moreover, the set of stable functions is dense.
VO AAG 27

Local Stability
A map germ f : (R m ,0) → (R n ,0) is stable iff for any deformation g of f
there is a point p in the neighborhood of 0 such that the germ of g at p is
equivalent to f.
If f is stable, then all germs of f are locally stable. The converse is not
true (e.g. extra condition on values to be distinct).
VO AAG 28

Infinitesimal Stability
Let D m,n be the set of germs of differentiable functions f : (R m , 0) →
(R n , 0). Let G m be the group of germs of automorphisms of (R m ,0).
Obviously G m × G n acts on D m,n but this is not enough.
Take f ∈ D m,n represented by a function defined on U ⊂ R m . For any
x 0 ∈ U, x → (f(x + x 0 ) − f(x 0 )) is also in D m,n . For fixed U, this defines
an action, and the derivative of this action at 0 does not depend on the
choice of U.
We say that f is infinitesimally stable iff the differential of the action above
(isomorphism and translation together) is surjective.
VO AAG 29

Example
m = n = 1. We may assume that f(x) = x d for some d > 0. Then
the image of the differential of right equivalence together with translation
is the ideal generated by f ′ (x), which is the vectorspace generated by x a ,
a ≥ d − 1. The ideal generated by left equivalence is the vectorspace
generated by x b , d | b.
It follows that x d is stable iff d = 1 or d = 2.
VO AAG 30

Algebraic Criterion
f ∈ D m,n is infinitesimally stable iff the the quotient of R[[x 1 ,...,x m ]] n
by the module generated by the partial derivatives and by f i e j , where f i is
the i-th coordinate of f and e j is the j-th unit vector, is generated by the
unit vectors over R.
Example (x, y) ↦→ (x 3 + xy, y).
VO AAG 31

Whitneys Theorem
Assume m = n = 2. Then f is locally stable iff f is nonsingular, or f is
isomrphic to (x,y) ↦→ (x 2 ,y) (a fold), or (x, y) ↦→ (x 3 + xy, y) (a pleat).
Assume dim(M) = dim(N) = 2 and M is compact. Then f is stable iff it
is locally stable at any point, and the set of critical values in N is a curve
with only ordinary nodes and cusps. The set of stable maps is dense.
VO AAG 32

Other Cases
m = 1, n = 2: locally stable iff nonsingular, stable iff image has only
ordinary double points.
m = 1,n > 2: stable iff immersion as a submanifold
n = 1: Morse theorem.
m = 2, n = 3: locally stable iff nonsingular or (x, y) ↦→ (xy, y, x 2 ). Stable
iff image has local equation x = 0 or xy = 0 or xyz = 0 or x 2 − y 2 z = 0.
m = n ≥ 9: stable maps are not dense!
VO AAG 33

Blowing Up
The blowup of the plane C 2 is an algebraic map p : Y → C 2 , where Y is a
manifold that can be covered by two open subsets isomorphic to C 2 . It has
the following properties:
p −1 (o) is a curve E ⊂ Y. Apart from that, p restricts to an isomorphism
Y − E → C 2 − {o}. The points of E correspond to tangent directions at o.
Any automorphism of C 2 that fixes o lifts to an automorphism of Y
preserving E. Even more, any analytic automorphism of the germ of C 2 at
o lifts to a local automorphism of a neighborhood of E in Y fixing E.
VO AAG 34

Total transform and proper transform
If C ⊂ C 2 is a curve containing o, then p −1 (C) has two components: the
proper transform ˜C and the exceptional curve E. The pullback of the
equation F of C factors into x m ˜f(x, ỹ), where ˜f is the local equation of
˜C, x is the equation of the exceptional curve, and m is the order of C at o.
The curve ˜C may have “new” singular points in E. The sum of all orders
at these new points is less than or equal to m.
Theorem. The procedure of blowing up singular points is finite.
VO AAG 35

Invariants and blowing up
The curve C is irreducible iff all proper transforms intersect E in only one
point.
For any plane curve singularity, the Milnor number µ is defined as the
intersection multiplicity of the two partial derivatives, and the δ-invariant is
defined as the height of the integral closure.
Assume that P 1 ,...,P k are the singularities of the proper transform lying
on E. Then the formulas hold
µ = m(m − 1) + µ 1 + · · · + µ k − k + 1
δ = m(m − 1)/2 + δ 1 + · · · + δ k
VO AAG 36

Point Blowups in Higher Dimension
The blowup of a point in C n is an algebraic map p : Y → C n , where Y is a
manifold that can be covered by n open subsets isomorphic to C n . It has
the following properties:
p −1 (o) is a hypersurface E ⊂ Y. Apart from that, p restricts to an
isomorphism Y − E → C n − {o}. The points of E correspond to tangent
directions at o.
VO AAG 37

Blowup of Submanifolds
Let o ∈ C n . Then the blowup the submanifold L = {o} × C m ⊂ C n+m is
just the direct product of the blowup of C n at o with C m . It can be covered
by n open subsets isomorphic to C n × C m . The exceptional hypersurface is
p −1 (L).
Example: blowup x = y = 0 in C 3 .
VO AAG 38

Transversality
Assume H : x = 0 is an exceptional divisor in C 3 . We blow up the curve
C : y = x + z 2 = 0.
Step 1: Coordinate transformation making the curve into a line (x is
replaced by x − z 2 ):
H : x − z 2 = 0, C : y = x = 0.
Step 2: First chart of blowup: x is replaced by xy.
exceptional divisor becomes singular.
Then the “old”
VO AAG 39

Transversality
Better: only blowup centers which intersect all old exceptional hypersurfaces
transversally. This means: for any point, there is an analytic coordinate
system such that the center is defined by coordinates AND all exceptional
hypersurfaces are defined by coordinates.
VO AAG 40

Variants of Resolution
Resolution of varieties: sequence of blowups such that the proper transform
is nonsingular.
Resolution of functions: sequence of blowups such that the pullback can
locally be defined by products of coordinates (monomials).
Resolution of ideals: sequence of blowups such that the pullback is locally
generated by such a monomial.
Resolution of hypersurfaces can be reduced to resolution of functions.
Resolution of varieties and resolution of functions can be reduced to
resolution of ideals.
VO AAG 41

Controlled Ideals, Singular Locus, Controlled Transform
A controlled ideal is an ideal together with a positive integer.
The singular locus of a controlled ideal (I,c) is the set of all points where
the ideal has order at least c.
If the center is contained in the singular locus, then the controlled transform
is defined by dividing out the c-th power of the exceptional ideal.
Resolution of controlled ideals: sequence of blowups with center in the
singular locus, such that the singular locus is empty.
VO AAG 42

Reduction Problems
Can you reduce the resolution problem of ideals to the resolution problem
of controlled ideals?
Conversely?
A controlled ideal is called tight iff its order is never greater then c. Can you
reduce the resolution problem of controlled ideals to the resolution problem
of tight controlled ideals?
VO AAG 43

Sum of Controlled Ideals
Let (I 1 ,c 1 ) and (I 2 ,c 2 ) be two controlled ideals. Then the sum is defined
as (I 3 , c 1 c 2 ), where I 3 = I c 2
1 · Ic 1
2 . This operation is commutative and
associative.
Proposition. The singular set of the sum is equal to the intersection. Sum
commutes with controlled transform.
VO AAG 44

Rees Algebras
Let R be the function ring of an algebraic or analytic variety (for instance,
R = C[x 1 , ...,x n ]). A Rees algebra over R is a graded ring A = ⊕ ∞ i=0 A i,
such that R = A 0 ⊇ A 1 ⊇ A 2 ⊇ . ...
The Rees algebra is finitely generated iff there exists b such that A is
generated by all elements of degree at most b.
The singular set of a Rees algebra is the set of all points where A i has order
at least i.
Consider a blowup at a center in the singular locus. Then the proper
transform of the Rees algebra is defined as the Rees algebra generated by
the controlled transforms of (A i ,i) in degree i.
VO AAG 45

Translation Controlled Ideals - Rees Algebras
Let (I,c) be a controlled ideal. Then the Rees algebra A(I,C) is defined
as the one generated by I in degree c.
Conversely, let A be a Rees algebra which is generated in degree at most b.
Then the associated controlled ideal is defined as the sum of the controlled
ideals (A i ,i), i = 1,...,b.
VO AAG 46