Calculations in Deformation Theory
Nathan Owen Ilten
UC Berkeley
February 22nd, 2012
Deformation Basics
Computing Versal Deformations
Deformation Basics
Computing Versal Deformations
The Setup
The Setup
I
Let S = C[x1 , . . . , xd ].
The Setup
I
Let S = C[x1 , . . . , xd ].
I
Consider polynomials f1 , f2 , . . . , fm ∈ S.
The Setup
I
Let S = C[x1 , . . . , xd ].
I
Consider polynomials f1 , f2 , . . . , fm ∈ S.
I
These generate an ideal I in S.
The Setup
I
Let S = C[x1 , . . . , xd ].
I
Consider polynomials f1 , f2 , . . . , fm ∈ S.
I
These generate an ideal I in S.
I
They also define an affine scheme X ⊂ Cd .
The Setup
I
Let S = C[x1 , . . . , xd ].
I
Consider polynomials f1 , f2 , . . . , fm ∈ S.
I
These generate an ideal I in S.
I
They also define an affine scheme X ⊂ Cd .
Basic idea: we deform X by perturbing the fi .
Example I: An A1 Singularity
Example I: An A1 Singularity
I
Take d = 3, f = x 2 + y 2 − z 2 .
Example I: An A1 Singularity
I
Take d = 3, f = x 2 + y 2 − z 2 .
Example I: An A1 Singularity
I
Take d = 3, f = x 2 + y 2 − z 2 .
I
Perturb this with a parameter t to get e
f = x 2 + y 2 − z 2 − t.
Example I: An A1 Singularity
I
Take d = 3, f = x 2 + y 2 − z 2 .
I
Perturb this with a parameter t to get e
f = x 2 + y 2 − z 2 − t.
I
This cuts out a scheme X ⊂ C3 × C with a natural projection
map π : X → C.
Example I: An A1 Singularity
I
Take d = 3, f = x 2 + y 2 − z 2 .
I
Perturb this with a parameter t to get e
f = x 2 + y 2 − z 2 − t.
I
This cuts out a scheme X ⊂ C3 × C with a natural projection
map π : X → C.
I
The fiber over 0 is just X . The fiber over t 6= 0 is smooth.
Example II: A Line in C3
Example II: A Line in C3
I
Take d = 3, f1 = x − y , f2 = x − z, f3 = y − z.
Example II: A Line in C3
I
Take d = 3, f1 = x − y , f2 = x − z, f3 = y − z.
I
These equations cut out a line x = y = z in C3 .
Example II: A Line in C3
I
Take d = 3, f1 = x − y , f2 = x − z, f3 = y − z.
I
These equations cut out a line x = y = z in C3 .
I
Now perturb:
fe1 = x − y + t
fe2 = x − z + t
fe3 = y − z + t
Example II: A Line in C3
I
Take d = 3, f1 = x − y , f2 = x − z, f3 = y − z.
I
These equations cut out a line x = y = z in C3 .
I
Now perturb:
fe1 = x − y + t
fe2 = x − z + t
fe3 = y − z + t
I
Something is fishy! The fiber over t 6= 0 is a point!
Example II: A Line in C3
I
Take d = 3, f1 = x − y , f2 = x − z, f3 = y − z.
I
These equations cut out a line x = y = z in C3 .
I
Now perturb:
fe1 = x − y + t
fe2 = x − z + t
fe3 = y − z + t
I
Something is fishy! The fiber over t 6= 0 is a point!
Problem: the relation f2 − f1 = f3 doesn’t lift to a relation among
the fei .
Lifting Relations
Lifting Relations
Consider the start of a free resolution of S/I :
R
F
· · · −−−−→ S n −−−−→ S m −−−−→ S −−−−→ S/I −−−−→ 0.
Here, F is a matrix whose columns are just the fi .
Lifting Relations
Consider the start of a free resolution of S/I :
R
F
· · · −−−−→ S n −−−−→ S m −−−−→ S −−−−→ S/I −−−−→ 0.
Here, F is a matrix whose columns are just the fi .
Consider a ring of deformation parameters T = C[t1 , . . . , te ] and
e = S ⊗ T.
set S
Lifting Relations
Consider the start of a free resolution of S/I :
R
F
· · · −−−−→ S n −−−−→ S m −−−−→ S −−−−→ S/I −−−−→ 0.
Here, F is a matrix whose columns are just the fi .
Consider a ring of deformation parameters T = C[t1 , . . . , te ] and
e = S ⊗ T.
set S
em → S
e be a perturbation of F .
Let Fe : S
Lifting Relations
Consider the start of a free resolution of S/I :
R
F
· · · −−−−→ S n −−−−→ S m −−−−→ S −−−−→ S/I −−−−→ 0.
Here, F is a matrix whose columns are just the fi .
Consider a ring of deformation parameters T = C[t1 , . . . , te ] and
e = S ⊗ T.
set S
em → S
e be a perturbation of F .
Let Fe : S
Definition
The relations R lift with respect to Fe subject to equations
e :S
en → S
e m restricting to R such
g1 , . . . , gk ∈ T if there exists a R
that
e ⊂ hgi i
Im Fe · R
.
Example III: Lifting Relations
Example III: Lifting Relations
I
Take d = 4 and consider the matrix
F =
x1 x3 − x22
x2 x4 − x32
x1 x4 − x2 x3
.
Example III: Lifting Relations
I
Take d = 4 and consider the matrix
F =
I
x1 x3 − x22
x2 x4 − x32
A relation matrix is given by


x4
x3
x1  .
R =  x2
−x3 −x2
x1 x4 − x2 x3
.
Example III: Lifting Relations
I
Take d = 4 and consider the matrix
Fe =
I
x1 x3 − x22 + tx2 x2 x4 − x32 − tx4 x1 x4 − x2 x3
A relation matrix is given by


x4
x3
x1  .
R =  x2
−x3 −x2
I
We can perturb F to Fe .
.
Example III: Lifting Relations
I
Take d = 4 and consider the matrix
Fe =
x1 x3 − x22 + tx2 x2 x4 − x32 − tx4 x1 x4 − x2 x3
I
A relation matrix is given by


x4
x3
e =  x2
.
x1
R
−x3 −x2 + t
I
We can perturb F to Fe .
I
e
A lifting of R is given by R.
.
Deformations of Affine Schemes
Deformations of Affine Schemes
Definition
A deformation of X ⊂ Cd over Z = Spec (T /hgi i) consists of a
perturbation Fe of F such that the relations lift with respect to Fe ,
subject to the equations gi .
Deformations of Affine Schemes
Definition
A deformation of X ⊂ Cd over Z = Spec (T /hgi i) consists of a
perturbation Fe of F such that the relations lift with respect to Fe ,
subject to the equations gi .
I
Fe defines a scheme X ⊂ Cd × Z and a map π : X → Z .
Deformations of Affine Schemes
Definition
A deformation of X ⊂ Cd over Z = Spec (T /hgi i) consists of a
perturbation Fe of F such that the relations lift with respect to Fe ,
subject to the equations gi .
I
Fe defines a scheme X ⊂ Cd × Z and a map π : X → Z .
I
X is the total space, Z the base space of the deformation.
Deformations of Affine Schemes
Definition
A deformation of X ⊂ Cd over Z = Spec (T /hgi i) consists of a
perturbation Fe of F such that the relations lift with respect to Fe ,
subject to the equations gi .
I
Fe defines a scheme X ⊂ Cd × Z and a map π : X → Z .
I
X is the total space, Z the base space of the deformation.
Example
For hypersurfaces, arbitrary perturbations are allowed.
Induced Deformations
Induced Deformations
I
Consider a deformation of X .
Induced Deformations
I
Consider a deformation of X .
I
We can induce other deformations of X by applying changes
of coordinates to the variables xi and substituting in new
deformation parameters for the ti .
Induced Deformations
I
Consider a deformation of X .
I
We can induce other deformations of X by applying changes
of coordinates to the variables xi and substituting in new
deformation parameters for the ti .
Example
Induced Deformations
I
Consider a deformation of X .
I
We can induce other deformations of X by applying changes
of coordinates to the variables xi and substituting in new
deformation parameters for the ti .
Example
I
Consider the deformation given by e
f = x 2 + y 2 − z 2 − t.
Induced Deformations
I
Consider a deformation of X .
I
We can induce other deformations of X by applying changes
of coordinates to the variables xi and substituting in new
deformation parameters for the ti .
Example
I
Consider the deformation given by e
f = x 2 + y 2 − z 2 − t.
I
Can we induce the deformation given by x 2 + y 2 − z 2 − sz?
Induced Deformations
I
Consider a deformation of X .
I
We can induce other deformations of X by applying changes
of coordinates to the variables xi and substituting in new
deformation parameters for the ti .
Example
I
Consider the deformation given by e
f = x 2 + y 2 − z 2 − t.
I
Can we induce the deformation given by x 2 + y 2 − z 2 − sz?
I
Yes! Substitute t = − 14 s 2 and take the change of coordinates
z 7→ (z + 21 s).
Versal Deformations
Versal Deformations
Definition
A deformation of X is called (formally) versal if any (infinitesimal)
deformation of X may be induced from it.
Versal Deformations
Definition
A deformation of X is called (formally) versal if any (infinitesimal)
deformation of X may be induced from it.
I
e to contain formal
If dim Sing X = 0 and we allow Fe and R
power series, then X has a formally versal deformation.
Versal Deformations
Definition
A deformation of X is called (formally) versal if any (infinitesimal)
deformation of X may be induced from it.
I
e to contain formal
If dim Sing X = 0 and we allow Fe and R
power series, then X has a formally versal deformation.
Q: How can we compute a versal deformation of X ?
Versal Deformations
Definition
A deformation of X is called (formally) versal if any (infinitesimal)
deformation of X may be induced from it.
I
e to contain formal
If dim Sing X = 0 and we allow Fe and R
power series, then X has a formally versal deformation.
Q: How can we compute a versal deformation of X ?
A: Using Macaulay2 and the package VersalDeformations.
Deformation Basics
Computing Versal Deformations
Basic Features of the Package VersalDeformations
Basic Features of the Package VersalDeformations
Input:
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Output:
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Output:
I
A basis for TX1 , the space of deformations over Spec C[t]/t 2 .
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Output:
I
A basis for TX1 , the space of deformations over Spec C[t]/t 2 .
I
A basis for TX2 , which contains obstructions to lifting
deformations.
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Output:
I
A basis for TX1 , the space of deformations over Spec C[t]/t 2 .
I
A basis for TX2 , which contains obstructions to lifting
deformations.
I
A formally versal deformation of X (more details later).
Basic Features of the Package VersalDeformations
Input: A matrix F containing the equations of X .
Output:
I
A basis for TX1 , the space of deformations over Spec C[t]/t 2 .
I
A basis for TX2 , which contains obstructions to lifting
deformations.
I
A formally versal deformation of X (more details later).
Basic approach: iteratively lift deformations in TX1 to larger and
larger base spaces.
Computational Example I: Our A1 Singularity
Computational Example I: Our A1 Singularity
I
F = ( x 2 + y 2 − z 2 ).
Computational Example I: Our A1 Singularity
I
F = ( x 2 + y 2 − z 2 ).
I
Any first order deformation can be induced from
F + t1 · 1 = ( x 2 + y 2 − z 2 + t1 ) with t12 = 0.
Output of Command “versalDeformation”
Output of Command “versalDeformation”
The command “versalDeformation” outputs four lists FL, RL, GL,
and CL where
Output of Command “versalDeformation”
The command “versalDeformation” outputs four lists FL, RL, GL,
and CL where
I
FL is a list of matrices whose sum is the perturbation matrix
Fe of a versal deformation.
Output of Command “versalDeformation”
The command “versalDeformation” outputs four lists FL, RL, GL,
and CL where
I
I
FL is a list of matrices whose sum is the perturbation matrix
Fe of a versal deformation.
e of R.
RL is a list of matrices whose sum is a lifting R
Output of Command “versalDeformation”
The command “versalDeformation” outputs four lists FL, RL, GL,
and CL where
I
I
I
FL is a list of matrices whose sum is the perturbation matrix
Fe of a versal deformation.
e of R.
RL is a list of matrices whose sum is a lifting R
GL is a list of matrices whose sum G contains the equations
cutting out the versal base space.
Output of Command “versalDeformation”
The command “versalDeformation” outputs four lists FL, RL, GL,
and CL where
I
I
I
FL is a list of matrices whose sum is the perturbation matrix
Fe of a versal deformation.
e of R.
RL is a list of matrices whose sum is a lifting R
GL is a list of matrices whose sum G contains the equations
cutting out the versal base space.
These matrices solve the “deformation equation”
e tr + C · G = 0
(Fe · R)
where C is the sum of the list CL.
Computational Example II: The Cone over the Rational
Normal Curve of Deg. 3
Computational Example II: The Cone over the Rational
Normal Curve of Deg. 3
Take
F =
x1 x3 − x22 x2 x4 − x32 x1 x4 − x2 x3
.
Computational Example II: The Cone over the Rational
Normal Curve of Deg. 4
Computational Example II: The Cone over the Rational
Normal Curve of Deg. 4




Take F to be the transpose of 



x1 x3 − x22
x2 x4 − x32
x3 x5 − x42
x1 x4 − x2 x3
x2 x5 − x3 x4
x1 x5 − x2 x5




.



Computational Example II: The Cone over the Rational
Normal Curve of Deg. 4




Take F to be the transpose of 



I
x1 x3 − x22
x2 x4 − x32
x3 x5 − x42
x1 x4 − x2 x3
x2 x5 − x3 x4
x1 x5 − x2 x5




.



The base space has two components, C3 and C meeting in a
point.
Total Spaces Over the Components
Total Spaces Over the Components
The components come from two ways of writing the equations of
X:
Total Spaces Over the Components
The components come from two ways of writing the equations of
X:
x1 x2 x3 x4
rk
≤1
x2 x3 x4 x5
Total Spaces Over the Components
The components come from two ways of writing the equations of
X:
x1 x2 x3 x4
rk
≤1
x2 x3 x4 x5

x1 x2 x3
rk  x2 x3 x4  ≤ 1
x3 x4 x5

Total Spaces Over the Components
The components come from two ways of writing the equations of
X:
x1
x2
x3
x4
rk
≤1
x2 + s1 x3 + s2 x4 + s3 x5

x1 x2 x3
rk  x2 x3 x4  ≤ 1
x3 x4 x5

Total Spaces Over the Components
The components come from two ways of writing the equations of
X:
x1
x2
x3
x4
rk
≤1
x2 + s1 x3 + s2 x4 + s3 x5

x1
x2
x3
rk  x2 x3 + s4 x4  ≤ 1
x3
x4
x5

Further Features of the Package VersalDeformations
Further Features of the Package VersalDeformations
I
Can calculate TX1 , TX2 , and normal modules for projective X
in good situations.
Further Features of the Package VersalDeformations
I
Can calculate TX1 , TX2 , and normal modules for projective X
in good situations.
I
Can calculate versal deformations for projective X .
Further Features of the Package VersalDeformations
I
Can calculate TX1 , TX2 , and normal modules for projective X
in good situations.
I
Can calculate versal deformations for projective X .
I
Can calculate local (multigraded) Hilbert schemes.
Further Features of the Package VersalDeformations
I
Can calculate TX1 , TX2 , and normal modules for projective X
in good situations.
I
Can calculate versal deformations for projective X .
I
Can calculate local (multigraded) Hilbert schemes.
I
Can lift deformations in given tangent direction to a
one-parameter family.
A Toric Fano Threefold
A Toric Fano Threefold
Let X be the projective subscheme of P8 cut out by
xi+1 xi−1 − xi y0
1≤i ≤6
y02
y02
1≤i ≤3
xi xi+3 −
y1 y2 −
where P8 has coordinates x1 , . . . , x6 , y0 , y1 , y2 .
A Toric Fano Threefold
Let X be the projective subscheme of P8 cut out by
xi+1 xi−1 − xi y0
1≤i ≤6
y02
y02
1≤i ≤3
xi xi+3 −
y1 y2 −
where P8 has coordinates x1 , . . . , x6 , y0 , y1 , y2 .
A Toric Fano Threefold (cont.)
A Toric Fano Threefold (cont.)
The versal base space of X has three components.
A Toric Fano Threefold (cont.)
The versal base space of X has three components. X admits
smoothings to three different kinds of Fano threefolds.
A Toric Fano Threefold (cont.)
The versal base space of X has three components. X admits
smoothings to three different kinds of Fano threefolds.
Similar calculations + lots of hard work can be used to classify all
smoothings of Gorenstein Fano toric threefolds of degree ≤ 12.
References
Jan Arthur Christophersen and Nathan Owen Ilten.
Toric degenerations of low degree Fano threefolds.
arXiv:1202.0510v1 [math.AG], 2012.
Nathan Owen Ilten.
VersalDeformations — a package for computing versal
deformations and local Hilbert schemes.
arXiv:1107.2416v1 [math.AG], 2011.
Jan Stevens.
Computing versal deformations.
Experiment. Math., 4(2):129–144, 1995.
First Order Deformations
First Order Deformations
Definition
Let TX1 be the set of isomorphism classes of deformations of X
with base space Spec C[t]/t 2 .
First Order Deformations
Definition
Let TX1 be the set of isomorphism classes of deformations of X
with base space Spec C[t]/t 2 .
TX1 may be computed as the cokernel of
J : S d → HomS (I , S/I ) ⊂ (S/I )m
∂fi
where J is the Jacobian matrix ∂x
.
j
ij
First Order Deformations
Definition
Let TX1 be the set of isomorphism classes of deformations of X
with base space Spec C[t]/t 2 .
TX1 may be computed as the cokernel of
J : S d → HomS (I , S/I ) ⊂ (S/I )m
∂fi
where J is the Jacobian matrix ∂x
.
j
ij
If dim Sing(X ) = 0, then dimC TX1 < ∞.
First Order Deformations (cont.)
First Order Deformations (cont.)
I
Choose φi ∈ Hom(S m , S) i = 1, . . . , e which represent a basis
of TX1 .
First Order Deformations (cont.)
I
Choose φi ∈ Hom(S m , S) i = 1, . . . , e which represent a basis
of TX1 .
I
Set T = C[t1 , . . . , te ] with maximal ideal m = ht1 , . . . , te i.
First Order Deformations (cont.)
I
Choose φi ∈ Hom(S m , S) i = 1, . . . , e which represent a basis
of TX1 .
I
Set T = C[t1 , . . . , te ] with maximal ideal m = ht1 , . . . , te i.
em → S
e be the perturbation of F = F 0 defined by
Let F 1 : S
I
1
0
F =F +
e
X
i=1
t i φi .
First Order Deformations (cont.)
I
Choose φi ∈ Hom(S m , S) i = 1, . . . , e which represent a basis
of TX1 .
I
Set T = C[t1 , . . . , te ] with maximal ideal m = ht1 , . . . , te i.
em → S
e be the perturbation of F = F 0 defined by
Let F 1 : S
I
1
0
F =F +
e
X
t i φi .
i=1
I
The relations R lift with respect to F 1 subject to m2 to some
en → S
em.
R1 : S
First Order Deformations (cont.)
I
Choose φi ∈ Hom(S m , S) i = 1, . . . , e which represent a basis
of TX1 .
I
Set T = C[t1 , . . . , te ] with maximal ideal m = ht1 , . . . , te i.
em → S
e be the perturbation of F = F 0 defined by
Let F 1 : S
I
1
0
F =F +
e
X
t i φi .
i=1
I
The relations R lift with respect to F 1 subject to m2 to some
en → S
e m . This can be computed using matrix quotients
R1 : S
in Macaulay2.
The Deformation Equation, Part I
The Deformation Equation, Part I
I
Goal: lift this deformation to a “larger” base space.
The Deformation Equation, Part I
I
I
Goal: lift this deformation to a “larger” base space.
e m , S),
e R i−1 ∈ Hom(S
en, S
e m ), we would
Given F i−1 ∈ Hom(S
i
i
like to find F and R such that
The Deformation Equation, Part I
I
I
Goal: lift this deformation to a “larger” base space.
e m , S),
e R i−1 ∈ Hom(S
en, S
e m ), we would
Given F i−1 ∈ Hom(S
i
i
like to find F and R such that
1. F i ≡ F i−1
mod mi , R i ≡ R i−1
mod mi ;
The Deformation Equation, Part I
I
I
Goal: lift this deformation to a “larger” base space.
e m , S),
e R i−1 ∈ Hom(S
en, S
e m ), we would
Given F i−1 ∈ Hom(S
i
i
like to find F and R such that
1. F i ≡ F i−1 mod mi , R i ≡ R i−1
2. F i · R i ≡ 0 mod mi+1 .
mod mi ;
The Deformation Equation, Part I
I
I
Goal: lift this deformation to a “larger” base space.
e m , S),
e R i−1 ∈ Hom(S
en, S
e m ), we would
Given F i−1 ∈ Hom(S
i
i
like to find F and R such that
1. F i ≡ F i−1 mod mi , R i ≡ R i−1
2. F i · R i ≡ 0 mod mi+1 .
In general, this is not possible!!!
mod mi ;
The Deformation Equation, Part II
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
I
Choose V ∈ Hom(S l , S n ) such that its columns represent a
basis of TX2 .
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
I
Choose V ∈ Hom(S l , S n ) such that its columns represent a
basis of TX2 .
e
It is possible to inductively construct F i , R i , G i−2 ∈ Hom(S,
l
i−2
l
n
e
e
e
S ), C
∈ Hom(S , S ) solving
I
(F i R i )tr + C i−2 G i−2 ≡ 0
such that:
mod mi+1 .
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
I
Choose V ∈ Hom(S l , S n ) such that its columns represent a
basis of TX2 .
e
It is possible to inductively construct F i , R i , G i−2 ∈ Hom(S,
l
i−2
l
n
e
e
e
S ), C
∈ Hom(S , S ) solving
I
(F i R i )tr + C i−2 G i−2 ≡ 0
mod mi+1 .
such that:
1. F i , R i , G i−2 , C i−2 reduce to F i−1 , R i−1 , G i−3 , C i−3 modulo
mi ;
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
I
Choose V ∈ Hom(S l , S n ) such that its columns represent a
basis of TX2 .
e
It is possible to inductively construct F i , R i , G i−2 ∈ Hom(S,
l
i−2
l
n
e
e
e
S ), C
∈ Hom(S , S ) solving
I
(F i R i )tr + C i−2 G i−2 ≡ 0
mod mi+1 .
such that:
1. F i , R i , G i−2 , C i−2 reduce to F i−1 , R i−1 , G i−3 , C i−3 modulo
mi ;
2. G i−2 and C i−2 vanish for i < 2;
The Deformation Equation, Part II
I
If dim Sing(X ) = 0, there is a finite dimensional C-vector
space TX2 containing obstructions to lifting F i , R i .
I
Choose V ∈ Hom(S l , S n ) such that its columns represent a
basis of TX2 .
e
It is possible to inductively construct F i , R i , G i−2 ∈ Hom(S,
l
i−2
l
n
e
e
e
S ), C
∈ Hom(S , S ) solving
I
(F i R i )tr + C i−2 G i−2 ≡ 0
mod mi+1 .
such that:
1. F i , R i , G i−2 , C i−2 reduce to F i−1 , R i−1 , G i−3 , C i−3 modulo
mi ;
2. G i−2 and C i−2 vanish for i < 2;
3. C 0 is of the form V · D, where D ∈ Hom(S d , S d ) is a
diagonal matrix.
The perturbation limi→∞ F i gives a formally versal deformation
over the base space cut out by the rows of limi→∞ G i−2 .