Deformation

1. Deformation of Singularities
We introduce a deformation of scheme Xo as a flat map π : X → S
with Xo ∼
= π −1 (0)
Definition 1.1. Let (X, x) and (S, s) be schemes. A deformation of
(X, x) over (S, s) consist of a flat morphism φ : (X, x) → (S, x) of scheme
together with an isomophism from (X, x) to the fibers of φ, (X, x) →
Xs := (φ−1 (s), x) represented as a Cartesian diagram
X
i
(X, x)
φ
{pt}
(S, s)
Remark 1.2. A morphism between two deformation π : (X, 0) → (S, 0)
and π 0 : (X 0 , x0 ) → (S, s) of Xo , 0 over the same base (S, 0) is a morphism
f : (X, x) → (X 0 , 0)
over (S, 0) i.e. π 0 ◦ h = π compatible with embedding i : (X, 0) → (X, 0)
and i0 : (X0 , 0) → (X 0 , 0)
Example 1.3. We consider the map π : P1 → P3 whose image describes
a quartic cone,Co ∈ P3 given by :
(x0 : x1 : x2 : x3 ) = (s41 − s42 : s31 s2 : s21 s22 : s1 s32 )
The curve Co is a complete intersection with equations
x22 − x1 x3 , x21 − xo x2 − x23 ,
with double point lying in (1 : 0 : 0 : 0). By permuting the map f → ft ,
we obtain in general a smooth rational quartic curve Ct .
Suppose we consider the local coordinates (x, y, z) and equations, y =
0, xz = 0. The curve germ (0, 0) is the image of the multigerm;
π : C ∪ C → C3
given by π(s1 ) → (s1 , 0, 0) and π(s2 ) = (0, 0, s2 ).
Now consider, the map;
F : (C ∪ C) × C → C3
defined as ;F (s1 , t) = (s1 , 0, 0, t) and F (s2 , t) = (0, t, s2 , t). For t 6= 0 we
obtain yx = zx = y 2 = zy = 0, describing the same plane curve with an
embedded point sticking in the y−direction.
1
2
Remark 1.4. The image X of F in C3 × C = C4 , consist of two planes
with one point in common,the simplest example of non-normal isolated
surface singularity.
Definition 1.5 (Infinitesimal deformation). An infinitesimal deformation of an affine variety X, is a flat morphism together with C−isomorphism
∼
XT × Spec C → X
such that it induces a commutative diagram
i
X
X̄
φ
Spec C
Spec R
Example 1.6. For n ≥ 1, we consider deformation of An surface singularities given by
C{x,y,z}
(x2 +y 2 +z n+1 )
O
Co
o
C{x,y,z,t}
(x2 +y 2 +z n+1 +t)
O
C {t} .
This singularity is deformed by constant t.
Definition 1.7. Let A = C[t]/(t2 ) be denoted by A = C ⊕ C[t] with
t2 = 0. The deformations of an affine scheme X over A are called first
order deformation. Suppose IA is an ideal of in A[x1 , x2 , . . . , xn ] then
IA = (f1 + g1 t), . . . , (fn−d + gn−d t)
where gi ∈ C[x1 , . . . , xn ].
Example 1.8. We consider a hypersurface X = V(fi ) ⊆ An . Then,
∂f
∂f SpecA = C[X]/ f,
,...,
∂x1
∂xn
2
Further, if we consider a cone, X = V (z − xy) ⊆ A3 , then we have that,
SpecA = C[x, y, z]/(f, −y, −x, 2z)
2
with f = z − xy + t.
n
Lemma 1.9. Let X = Xn,q be cyclic quotient with n−q
= [a , . . . , ae−1 ]
P 2
0
and suppose e > 3. The dimension of TX is (e − 4) + (a − 1), and
basis is given by the vector field;
1
1
∂
∂ ϑ(a)
j
x
−
i
y
; 1 ≤ a ≤ a − 1 ≤ ≤ e − 1
= −
j
a
i
n (x y )
∂x
∂y
3
1 1
∂
∂ (j
−
j
)x
−
(i
−
i
)y
; 3≤≤e−2
−1
−1
n xi y j ∂x
∂y
To compute the action on the equation,
field in
P we express ∂the vector
P
∂
=
iρ zρ ∂ρ and y ∂y =
jρ zρ ∂ρ , we
terms of the ∂ρ := ∂∂ ρ from x ∂x
obtain;
1 1 X
ϑ(a)
=
−
(iρ j − i jρ )zρ ∂ρ
n za
1 1 X
δ = − a
iρ (j − j−1 ) − (i − i−1 )jρ zρ ∂ρ
n z
with a i = i−1 + i+1 . We compute that i−1 j − i j−1 is independent
of so equal to, io j1 − i1 jo = n. Therefore
1
ϑ(a)
= (a) . . . − z−1 ∂−1 + z+1 ∂+1 + . . .
z
,
1
δ = − c . . . + z−1 ∂−1 + z ∂ + (a − 1)z+1 ∂+1 + . . .
z
P
It is sufficient to give two coefficients ci of the vector fields ci zi ∂i above
because , the ci satisfy the equation ai ci = ci−1 + ci+1 .
Arndt has proven that an infinitesimal deformation of X is completely
determined by its action on the e−2 equations z−1 z+1 −za , on the vector
(a)
space TX0 we take the coordinates t , 1 ≤ a ≤ a−1 , = 2, . . . , e − 1 and
s , = 3, . . . , e − 1. We write the infitesimal deformation by as:
δ = −
a −2
z1 (z+1 − s+1 ) − (z − s )(za −1 + t(1)
+ . . . + ta −1 )
z
To make this formula valid for all we introduce dummy variables,
s2 , se−1 and se whose value we set equal to zero. Of we lift the vector
(a)
fields ϑ and δ to C o (NX ) by taking ρ ≥ on Ur+1 , we obtain exactly
ϑ(a)
=
∂
, δ =
(a)
∂
∂s
∂t
For e = 3 one has to modify the previous formula.
The versal deformation of An−1 is given as;
z1 z3 − (z2n + t(2) z2n−2 + . . . + tn−1 z2 + tn )
and
∂
∂
1 1
∂ =−
x
−y
, 2 ≤ a ≤ n.
(a)
a
∂t
n (xy)
∂x
∂y
Theorem 1.10. The first order deformations of X are in one-one correspondence with the first cohomology of the tangent space ,TX1 .
4
Corollary 1.11. The set of deformation X over A0 = A ⊕ C which
extends deformation over A are in one-to -one correspondence with first
order deformation of X
Correspondence between MCKAY quiver and Cyclic
quotient Singularities
For the cyclic quotient singularity, An − case, we let the group Zn1 =
Z/(n + 1)Z, n ≥ 1 act on C[u, v] by :
−1
u 7→ ζn+1 u, v 7→ ζn+1
v
(1.1)
with ζn+1 is a primitive (n + 1) of root of unity. We obtain the generating
invariants xo = un+1 , x1 = uv, x2 = v n+1 , and the generating equation
xo x2 = (uv)n+1 = xn+1
1
The MACKAY- quiver is of the form
DIAGRAM With n+1 vertices are one dimensional representations define
by tensoring ;
j
Mj : u → ζn+1
u, u ∈ C, j = 0, . . . , n
This representation 1.1 yields :
Mj ⊗ C2 = Mj−1 ⊕ Mj+1 , j ∈ Zn+1 ,
We fix a basis for each Mj with a number uj and


0 0··· ···0
0
un
uo 0 · · · · · · 0
0
0


0 u1 · · · · · · 0
0
0
U =
.
..
..
..
.. 
 ..
.
.
.
.
0 0 · · · · · · un−1 0
Such that, we can identify Mj → Mj+1 as a linear homomorphism
by describing the endomorphism U ∈ End M , with expression M =
Mo ⊕ . . . ⊕ Mn
Similarly,we have an endomorphism V ∈ End M with


0 v1 0 · · · 0
 0 0 v2 · · · 0 
. .
..
..
.. 
V =
.
.
.
 .. ..

0 0 ··· 0 v 
n
v0
0
···
0
0
These endomorphism have diagonal:
U n+1 = uo · · · un En+1 , U V = diag(vo un ·v1 uo · · · vn un−1 ), V n+1 = v0 · · · vn En+1
5
Taking the entries of this map; uo · · · un , vo un , v1 uo , . . . , vn un−1 , vo . . . , un
as generators of subalgebra of C[uo , . . . , un vo . . . , vn ], presented by
(o)
(n)
(n)
C[xo , x1 , . . . , x1 , x2 ]/hxo x2 = xo1 . . . x1 i
(j)
with xo = uo . . . un , x2 = vo . . . vn , x1 = vj uj−1 , j ∈ Zn+1 We express
the corresponding deformation in the form :
n
X
[U, V ] = U V − V U = diag(λ, . . . , λn ),
λj = 0
j=0
The following theorem RO98 says that in order to understand the
deformation theory of cyclic quotient surface singularities we need to
consider underlying quiver seriously as MCKAY- quiver.
Theorem 1.12. For a given (n, q) take in C[uo , . . . , un−1 , vo , . . . , vn−1 ]
the subalgebra generated by those elements of diagonal matrices
U iρ V jρ , ρ = 0, . . . r + 1
belonging to a special representation. then this algebra is canonically
isomorphic to the algebra of the total space of the Artin components of
the An,q singularity upto smooth factor.
6
References
[R098] Riemenshneider Oswal,Cyclic quotient Singularities. Constructing the
Artin Components via the MCKAY-Quiver.Singularity and complex Analytic
Geometry,,1998.