§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Obstructions to deforming curves lying on a K3
surface in a Fano 3-fold
Hirokazu Nasu
Tokai University
March 3, 2016,
Seminar in Algebra and Algebraic Geometry
@University of Oslo
(based on arXiv:1601.07301)
1 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
§1 Introduction (Motivation)
2 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Hilbert scheme
We work over a field k = k with char k ≥ 0.
V ⊂ P n: a closed subscheme.
OV (1): a very ample line bundle on V .
X ⊂ V : a closed subscheme.
P = P(X) = χ(X, O X (n)): the Hilbert polynomial of X.
Theorem 1.1 (Grothendieck’60)
There exists a projective scheme H (called the Hilbert scheme of V ),
parametrizing all closed subschemes X′ of V with the same Hilbert
polynomial P as X.
3 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Hilbert scheme of smooth connected curves
We consider the Hilbert scheme of smooth connected curves C ⊂ V .
Let d and g denote the degree d(C) := deg OC (1) and arithmetic genus
g(C) of C.
Notation 1.2
Hilb V
∪
= the (full) Hilbert scheme of V
open
Hilb sc V :
∪
= {smooth connected curves C ⊂ V}
Hilb sc
V:
d,g
= {curves of degree degree d and genus g}
closed
open
4 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
The Hilbert scheme of space curve Hilb sc P3
The space curves have been classically studied by many algebraic
geometers, e.g. Noether and Halphen. The problem of classifying space
curves is nowaday reduced to the study of the Hilbert scheme.
Why do we study Hilb sc P3 ? Some reasons are:
For every smooth curve C, there exists a curve C′ ⊂ P3 s.t. C′ ≃ C.
Recently, the classification of space curves has been applied to the
study of birational automorphism
Φ : P3 d P3
(for the construction of Sarkisov links in [Blanc-Lamy,2012]).
5 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Several results on Hilb sc P3
Let H d,g denote the subscheme Hilb sc
P3 . (Hilb sc P3 =
d,g
⊔
H d,g .)
If g ≤ d − 3, then H d,g is irreducible [Ein,’86] and H d,g is generically
smooth of expected dimension 4d.
(36)
In general, H d,g can become reducible, e.g H9,10 = W
1
[Noether].
⊔ W2(36)
the Hilbert scheme of arith. Cohen-Macaulay (ACM, for short)
curves are smooth [Ellingsrud, ’75].
def
C ⊂ P3 : ACM ⇐⇒ H1 (P3 , IC (l)) = 0 for all l ∈ Z
H d, g can have many generically non-reduced irreducible
components. [Mumford’62]
6 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Infinitesimal property of the Hilbert scheme
V : a smooth projective variety over k
X ⊂ V : a closed subscheme of V
I X (resp. N X/V ): the ideal (resp. normal) sheaf of X in V
Fact 1.3 (Tangent space and Obstruction group)
1
2
The tangent space of Hilb V at [X] is isomorphic to
Hom(I X , O X ) ≃ H0 (X, N X/V )
The group H1 (X, N X/V ) (⊂ Ext1 (I X , O X )) contains every
“obstruction” (explained later) to deforming X in V , if X is a locally
complete intersection in V .
7 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Let X be a locally complete intersection closed subscheme of V . Then
Fact 1.4
1
We have
h0 (X, N X/V ) − h1 (X, N X/V ) ≤ dim[X] Hilb V ≤ h0 (X, N X/V ).
2
In particular, if H1 (X, N X/V ) = 0, then Hilb V is nonsingular at [X] of
dimension h0 (X, N X/V ).
In general, we have H1 (X, N X/V ) , 0 (cf. Mumford’s ex.).
8 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Mumford’s example (a pathology)
S ⊂ P3 : a smooth cubic surface (≃ Blow6 pts P2 )
h = S ∩ P2 : a hyperplane section
E: a line on S
A general member of the complete linear system
Λ := |4h + 2E|
on S is a smooth connected curve C of degree 14 and genus 24. Such
curves C ⊂ P3 (with a variable S) are parametrised by a locally closed
irreducible subset
sc
P3
W = W (56) ⊂ Hilb14,24
of the Hilbert scheme. (dim |OP3 (3)| = 19, dim |OS (C)| = 37)
9 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
We compute that
h0 (NC/P3 ) = 56 + h1 (NC/P3 ) = 56 + 1 = 57.
By
56 = dim W ≤ dim[C] Hilb sc P3 ≤ h0 (NC/P3 ) = 57.
We now have a dichotomy between (A) and (B):
A
W is an irreducible component of (Hilb sc P3 )red . Hilb sc P3 is
generically non-reduced along W (i.e., non-reduced at the generic
point).
B
There exists an irreducible component W ′ ⫌ W . Hilb sc P3 is
generically smooth along W (i.e., smooth at the generic point).
Which is correct? { The answer is (A).
10 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
History
Later many non-reduced components of Hilb sc P3 were found by
Kleppe[’85], Ellia[’87], Gruson-Peskine[’82], Fløystad[’93], N[’05], and
more recently by Kleppe-Ottem[’15].
Moreover, to the question ”How bad can the deformation space of an
object be?”, Vakil[’06] has answered that
Law 1.5 (Murphy’s law in algebraic geometry)
Unless there is some a priori reason otherwise, the deformation space
may be as bad as possible.
11 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
A naive question
Nowadays non-reduced components of Hilbert schemes are not seldom.
However,
Question 1.6
What is/are the most important reason(s) (if any) for their existence?
Our answer is the following: (at least in Mumford’s ex.) a (−1)-curve E
(i.e. E ≃ P1 , E2 = −1) on the (cubic) surface is the most important.
12 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
A generalization of Mumford’s ex.
Theorem 1.7 (Mukai-N’09)
V : a smooth projective 3-fold. Suppose that
∃
E ≃ P1 ⊂ V and N E/V : globally generated,
∃
2
S: smooth surface s.t. E ⊂ S ⊂ V with (E2 )S = −1 and
H1 (NS/V ) = pg (S) = 0.
Then the Hilbert scheme Hilb sc V has infinitely many generically
1
non-reduced components.
Remark 1.8
1
2
V = P3 , S: a smooth cubic, E: a line. (Mumford’s ex.)
Many uniruled 3-folds V , e.g. Fano 3-folds (of index 2), P1 -bundles,
etc., satisfy the assumption of the theorem.
13 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Fano manifold and index
Definition 1.9
A projective manifold V with ample −KV is called Fano manifold.
Given a Fano manifold V , Fano index r(V) of V is defined by
{
r(V) := max k ∈ N
}
−KV = kH for some divisor H on V
A Fano manifold V of dimension n is called del Pezzo if
r(V) = n − 1, and called Mukai if r(V) = n − 2.
14 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Fano 3-folds of index 2 (del Pezzo 3-folds)
There exists a classification due to [Fujita,’80,’81] and [Iskovskih,’77].
del Pezzo 3-folds
V1
V2
V3
V4
=
=
=
=
(6) ⊂ P(3, 2, 1, 1, 1)
(4) ⊂ P(2, 1, 1, 1, 1)
(3) ⊂ P4
(2) ∩ (2) ⊂ P5
Plücker
V5 = [Gr(2, 5) ,→ P9 ] ∩ P6
Segre
V6 = P × P × P ,→ P
1
V6′
V7
1
1
Segre
7
= [P × P ,→ P ] ∩ P
= Blowpt P3 ⊂ P8
V8 = P
2
2
3 Veronese
,→
P9
8
n
1
2
3
4
weighted hypsurf. of deg. 6
weighted hypsurf. of deg. 4
cubic 3-fold
c. i. of two quadrics
5
linear sect. of Grass.
6
7
6
7
blow-up of P3 at a pt.
8
Veronese image of P3
15 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Good lines and Bad lines
V : a smooth del Pezzo 3-fold of degree n.
def
E ⊂ V : a line (⇐⇒ E ≃ P1 and deg O E (1) = 1).
Definition 1.10
A line E on V is called good if N E/V is trivial, and called bad, otherwise
( N E/V ≃ OP1 (k) ⊕ OP1 (−k) for k > 0).
If n , 8 (i.e., V , P3 ), then V contains a good line [Iskovskih,’77].
Theorem 1.11 (N,’10)
The closure W in Hilb sc V of the set



C ⊂ ∃ S: smooth hyp. sect. of V , and 




W := 



∃
C ⊂ V
C ∼ −KV S + 2E for E: good line
becomes a generically non-reduced component of Hilb sc V .
16 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Thus for every smooth del Pezzo 3-fold V , the Hilbert scheme Hilb sc V
contains a generically non-reduced component, whose general member
is contained in a smooth hyperplane section S of V (i.e., a del Pezzo
surface S).
The following question is due to Eiichi Sato (Kyushu univ.).
Question 1.12 (E. Sato’07)
For every smooth Mukai 3-fold V , does Hilb sc V contain a generically
non-reduced component?
A smooth hyperplane section of Mukai 3-fold is a K3 surface.
Example 1.13
A smooth quartic hypersurface V4 ⊂ P4 is a Mukai 3-fold. A smooth
hyperplane section of V4 is a quartic K3 surface S4 ⊂ P3 .
17 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§1.1 The Hilbert Scheme of Space Curves
§1.2 Infinitesimal Property of the Hilbert Scheme
§1.3 Mumford’s Example
Fano 3-folds of index 1 (Mukai 3-folds)
If ρ = 1, then we have the following classification.
Mukai 3-folds
V2 → P3 : a double cover
V4 = (4) ⊂ P
V6 = (2) ∩ (3) ⊂ P5
V8 = (2) ∩ (2) ∩ (2) ⊂ P6
4
Plücker
V10 = [Gr(2, 5) ,→ P9 ] ∩ P7 ∩ (2)
V12 ⊂ P8
V14
V16
V18
V22
=
⊂
⊂
⊂
Plücker
[Gr(2, 6) ,→ P14 ] ∩ P9
P10
P11
P13
−KV3
2
4
6
8
branched along (6) ⊂ P3
a quartic 3-fold
c. i. of a quadric and a cubic
c. i. of three quadrics
10
12
quadratic section of Grass.
14
16
18
22
linear section of Grass.
Umemura-Mukai 3-fold
18 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
§2 Infinitesimal analysis of the Hilbert scheme
19 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Generalizing Mumford’s example, we develop a technique to compute the
obstructions to lift a first order deformation of a curve on a 3-fold to a
second order deformation (called “Obstructedness Theorem”).
Setting:
V : a uniruled 3-fold
S: a surface
E: (−1)-curve on S
C: a curve on S
with C, E ⊂ S ⊂ V
Obst. Theorem
=⇒
Non-reduced
components of
Hilb sc V
We have recently obtained a generalization of this theorem.
20 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
First order deformation
V : a smooth projective variety over k.
X ⊂ V : a closed subscheme of V .
D = k[t]/(t 2 ): the ring of dual number.
Definition 2.1
A first order (infinitesimal) deformation of X in V is a closed subscheme
X′ ⊂ V × Spec D, flat over D, with X0 = X.
By the universal property of the Hilbert scheme, there exists a one-to-one
correspondence between
1
2
3
D-valued pts Spec D → Hilb V sending 0 7→ [X].
First order deformations X̃ of X in V
Global sections α of N X/V
21 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Applying the infinitesimal lifting property of smoothness to the surjection
k[t]/(t n+2 ) → k[t]/(t n+1 ) → 0,
of the artinian rings, we have
Proposition 2.2
If Hilb V is nonsingular at [X], then, for every n ≥ 1, every n-th order
deformation of X in V lifts to a (n + 1)-th order deformation of X in V .
In particular, if there exists a first order deformation X̃ of X not liftable to
any second order deformation, then Hilb V is singular at [X].
Definition 2.3
We say that X is unobstructed in V if Hilb V is nonsingular at [X], and
otherwise, we say X is obstructed in V .
22 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Obstructions and Cup products
Let X̃ ⊂ V × Spec k[t]/(t 2 ) be a first order deformation of X in V and α
the corresponding global section of N X/V . Let ob(α) denote the cup
product
ob(α) := α ∪ e ∪ α
∈ Ext1 (I X , O X ),
where e ∈ Ext1 (O X , I X ) is the extension class of an exact sequence
0 −→ I X −→ OV −→ O X −→ 0. Then
Theorem 2.4
If X is a locally complete intersection in V , then
1
2
ob(α) is contained in H1 (N X/V ) ⊂ Ext1 (I X , O X ).
ob(α) = 0 if and only if X̃ lifts to a second order deformation
X̃˜ ⊂ V × Spec k[t]/(t 3 ).
Here ob(α) (=: ob( X̃)) is called the primary obstruction of α (or X̃).
23 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Obstructions to deforming hypersurface and d-map
S ⊂ V : a smooth hypersurface of V .
Hilb cd V : the Hilbert scheme of Cartier divisor on V .
There exists a “Hilbert-Picard morphism”
ψS : Hilb cd V −→ Pic S,
S′ ⊂ V 7−→ OS (S′ ) = OV (S′ ) ⊗ OS .
(We have ψS (S) = OS (S) ≃ NS/V ). The tangent map
dS : H0 (S, NS/V ) −→ H1 (S, OS )
of ψS at [S] is called d-map for S ⊂ V .
Proposition 2.5
Let S̃ (or β ∈ H0 (NS/V )) be a first order deformation of S. Then
1
2
dS (β) ∈ H1 (OS ) represents the isomorphism class of L = NS̃/V S .
∪
ob(β) = dS (β) ∪ β, where H1 (S, OS ) × H0 (S, NS/V ) −→ H1 (S, NS/V ).
24 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Infinitesimal deformation with pole of hypersurface
Let E be an effective Cartier divisor on S.
Definition 2.6
A rational section β of NS/V admitting a pole along E of order m ( m ≥ 1),
i.e.,
β ∈ H0 (NS/V (mE)) \ H0 (NS/V ((m − 1)E))
is called an infinitesimal deformation with a pole of S ⊂ V along E (of
order m).
Put S◦ := S \ E and V ◦ := V \ E (open varieties).
Remark 2.7
By a natural map H0 (NS/V (mE)) ,→ H0 (NS◦ /V ◦ ), every infinitesimal
deformation with a pole induces a first order deformation of S◦ in V ◦ .
25 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Infinitesimal deformation with pole of invertible sheaf
We assume that for all m ≥ 1, the natural map
H1 (S, OS (mE)) ,→ H1 (S, OS ((m + 1)E))
is injective. Then there exists a natural filtration
H1 (OS (E)) ⊂ H1 (OS (2E)) ⊂ · · · ⊂ H1 (S◦ , OS◦ ).
on H1 (S◦ , OS◦ ).
Remark 2.8
Let L be an invertible sheaf on S. Every element of H1 (OS (mE)) is
regarded as the isomorphism class of an infinitesimal deformation L of L
with a pole along E.
26 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Polar d-map
The d-maps are extended to a version with a pole (i.e. polar d-maps).
Let dS (resp. dS◦ ) be the d-map for S ⊂ V (resp. S◦ ⊂ V ◦ ). Then by
Lemma 2.9 (Key Lemma1)
dS◦ (H0 (S, NS/V (mE))) ⊂ H1 (OS ((m + 1)E)).
we define a polar d-map
H0 (S, NS/V (mE)) −→
H1 (S, OS ((m + 1)E))
β
∈
∈
dS :
7−→
.
dS (β)
Then we consider the restriction to E (the principal part along E)
dS (β) E ∈ H1 (E, O E (mE))
of the image dS (β).
27 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
There exists an exact sequence
[0 −→ N E/S −→ N E/V −→ NS/V E → 0] ⊗ O E (mE).
|{z}
(2.1)
≃O E (E)
Let ∂ E denotes its coboundary
map. Then by the following lemma, the
principal part dS (β) E coincides with the coboundary ∂ E (β E ) of the above
exact sequence up to the scale m.
Lemma 2.10 (Key Lemma2)
The following diagram is commutative.
dS
H0 (S, N
−−−−−→
S/V (mE))


y|E
m∂ E
H0 (E, NS/V (mE) E ) −−−−−→
H1 (S, OS (mE))


y|E
1
H (E, O E (mE)).
28 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
A strategy in proving ob(α) , 0
In what follows, we assume that dim V = 3 and C ⊂ S ⊂ V is a sequence
of a smooth curve C and a smooth surface S in V .
Let α be a global section of NC/V . Since it is DIFFICULT to prove
ob(α) , 0 directly, we prove instead the nonzero of the IMAGE
H1 (πC/S (ob(α))) ∈ H1 (C, NS/V C ),
of ob(α) by the projection πC/S : NC/V → NS/V C (a reduction).
Definition 2.11
Define the exterior component of α and ob(α), respectively by
πC/S (α) := H0 (πC/S )(α)
and
obS (α) := H1 (πC/S )(ob(α)),
29 / 50
§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
A generalization of obstructedness theorem
Suppose that the exterior component πC/S (α) of α lifts
to an infinitesimal
deformation β ∈ H0 (NS/V (mE)) with pole, and let β E and ∂ E (β E ) be as
in the following diagram.
α


y
β
C
res
←−[
∃
res
β
7−→
∋
H0 (NC/V ) ∋


yπC/S
0
H (NS/V C ) ∋
∩
H0 (NS/V (mE)C )
res
←−
H0 (NS/V (mE))
β E


y
∂ E (β E )
H0 (N E/V
(mE))



yπE/S (mE)
∈ H0 (NS/V
(mE) E )



y∂E
∈ H1 (O E ((m + 1)E))
Figure: Relation between α, β E and ∂ E (β E )
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Theorem 2.12 (Main Theorem 1)
Let C ⊂ S ⊂ V , E, α, β be as above. The primary obstruction ob(α) of α
is nonzero, if
1
∆ := C + KV S − 2mE. The restriction map H0 (S, ∆) → H0 (E, ∆ E ) is
surjective.
2
m∂ E (β E ) ∪ β E , 0, where ∪ is taken by
∪
H1 (O E ((m+1)E))×H0 (NS/V (mE−C) E ) −→ H1 (NS/V ((2m+1)E−C) E ).
Remark 2.13
1
2
In [Mukai-N,’09], E2 < 0 and m = 1 were assumed. We replace the
conditions by H1 (OS (mE)) ,→ H1 (OS ((m + 1)E))
(∀ m ≥ 0)
E is not a (−1)-curve in general. If S is K3 surf., E is a (−2)-curve or
elliptic curve ( E2 = 0) (See §.3).
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§2.1 Obstructions and Cup Products
§2.2 Infinitesimal Deformation with Pole
§2.3 Obstructedness Theorem
Outline of Proof
We consider the image obS (α) of obS (α) by the natural surjection
H1 (C, NS/V C ) → H1 (C, NS/V ((m + 1)E)C )
and we prove obS (α) , 0 by the diagram
obS (α)
∥
|C
dS (β)C ∪ πC/S (α) ←−−−−−
|E
dS (β) ∪ β −−−−−→
dS (β) E ∪ β E
∥by Key Lemma
m∂ E (β E ) ∪ β E
□
, 0.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
§3 Obstructions to deforming curves lying on a
K3 surface
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Application
As an application, we study the deformations of curves C lying on a
smooth K3 surface S in a smooth Fano 3-fold V .
Definition 3.1
A smooth projective surface S is called a K3 surface, if KS ∼ 0 and
H1 (S, OS ) = 0.
A smooth quartic surface S4 ⊂ P3 is well known to be a K3 surface.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
In Mumford’s example, a general member C of the generically
non-reduced component W of Hilb sc P3 is contained in a smooth cubic
surface S3 ⊂ P3 and a (−1)-curve E (a line) on S3 plays an important role
in showing the non-reducedness of W (the obstructedness of C in P3 ).
One can naturally ask...
Question 3.2
What about curves C ⊂ P3 lying on a smooth quartic S4 ⊂ P3 ?
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
A result by Kleppe and Ottem
Theorem 3.3 (Kleppe-Ottem,’15)
Let S ⊂ P3 be a smooth quartic surface containing a line E, and such that
Pic S ≃ Zh ⊕ ZE, where h ∼ OS (1). If C ⊂ S is a smooth connected
curve of degree d > 16 and genus g, not a complete intersection in S,
and satisfies that D := C − 4h ≥ 0, then the 4-maximal family
W ⊂ Hilb sc
P3 containing C becomes
d,g
1
2
a generically smooth component of Hilb sc P3 if C.E ≥ 3,
a generically non-reduced component
of Hilb sc P3 if C.E ≤} 2 and
{
C / 4h + E, d ≥ 21 and g > min G(d, 5) − 1, d2 /10 + 21 .
Here G(d, 5) denotes the maximum genus of curves of degree d not
contained in a surface of degree 4. In ⃝
2 , the assumption on the genus g
is almost always satisfied if C.E ≤ 2.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Naive expectation and substitute for (−1)-curves
A smooth curve E on a K3 surface is rational iff E2 = −2.
One may expect...
Expectation 3.4
On K3 surface S, (-2)-curves E ≃ P1 on S control the deformations
(obstructedness) of an each curve C ⊂ S.
However, on K3 surfaces,
1
(−2)-curves E, and
2
elliptic curves F (then F2 = 0)
somehow control the deformation of C ⊂ S in V . We will see they play a
role very similar to that of a (−1)-curve on a smooth cubic surface
S3 ⊂ V3 in Mumford’s example!
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Settings
V : a smooth Fano 3-fold
S ⊂ V : a smooth K3 surface
C ⊂ S: a smooth connected curve of genus g
Since KS ∼ 0, by adjunction, we get
NS/V ≃ −KV S
and
NC/S ≃ KC .
This implies
H1 (S, NS/V ) = 0
and
H1 (C, NC/S ) ≃ k.
Thus S is unobstructed in V and C has a small obstruction group in S.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
A technical assumption
In what follows, we assume that
Assumption 3.5 (techinical assumption)
Assume that there exists a first order deformation S̃ of S which does not
contain any first order deformations C̃ of C.
This is due to the following...
Lemma 3.6
Under the technical assumption, the Hilbert-flag
scheme HF V becomes
nonsingular of expected dimension (KV S )2 /2 + g + 1 at (C, S).
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Hilbert-flag scheme and Divisor D
Definition 3.7
Given a projective scheme V , there exists a projective scheme HF V
(called the Hilbert-flag scheme of V ), which parametrises all the pairs
(C, S) of curves C and surface S with C ⊂ S ⊂ V . (cf. [Kleppe,’85]).
We denote the tangent space and the obstruction space of HF V at (C, S)
by Ai (C, S) (i = 0, 1), respectively. We also define a divisor D on S by
NS/V (−C) ∼ −KV S − C ∼ − D.
Then A2 (C, S) = 0 by the technical assumption, and there exists an exact
sequence
p1
0 −→ H0 (S, −D) −→ A1 (C, S) −→ H0 (C, NC/V ) −→ H1 (S, −D) −→ 0.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
First observation
Here the map p1 is the tangent map of the natural projection (a forgetful
morphism)
pr1 : HF V −→ Hilb V,
(C, S) 7−→ [C].
By the Serre duality, H1 ( D) = H1 (−D), and if H1 (D) = 0 then pr1 is
smooth at (C, S). Thus we get
Proposition 3.8
If H1 (S, D) = 0, then C is unobstructed.
Fact 1
If D is an effective divisor on a K3 surface S, and there exist no (−2)
curves and no elliptic curves on S, then H1 (S, D) = 0. (In fact,
Amp(S) = NE(S))
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Sharp Vanishing Theorem on K3 Surf.
For the non-vanishing of H1 (S, D) on a K3 surface S, we have
Lemma 3.9 (cf. Knutsen-Lopez’07)
Let S be a K3 surface, D a divisor. Suppose that D ≥ 0 and D2 ≥ 0.
Then H1 (S, D) , 0 if and only if
1
There exists a divisor ∆ ≥ 0 s.t. ∆2 = −2 and D.∆ ≤ −2, or
2
D ∼ nE for a nef and primitive divisor E > 0 with E2 = 0 and n ≥ 2.
Thus if there exists a (−2)-curve E ≃ P1 on S with D.E ≤ −2 or, if there
exists an elliptic curve E on S with D = mE for m ≥ 2, then H1 (D) is
nonzero.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Theorem 3.10 (Main Theorem 2)
Let C, S, V be as above. Put D := C + KV S and suppose D ≥ 0.
1
2
If there exist no (−2)-curves and no elliptic curves on S, or if
H1 (S, D) = 0, then C is unobstructed.
If D2 ≥ 0 and there exists a (−2)-curve E on S such that E.D = −2
and H1 (S, D − 3E) = 0, then we have h1 (S, D) = 1. If moreover,
π E/S (E) : H0 (E, N E/V (E)) −→ H0 (E, NS/V (E) E )
(♡)
is not surjective, then C is obstructed.
3
If there exists an elliptic curve F on S such that D ∼ mF for m ≥ 2,
then we have h1 (S, D) = m − 1. If moreover, π F/S (F) is not
surjective, then C is obstructed.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
In the theorem, the map (♡) is the map on the cohomology groups by the
projection
π E/S
[N E/V −→ NS/V E ] ⊗ O E (E).
Thus C is unobstructed in V if H1 (S, D) = 0. By using Main Theorem 1,
we partially prove that C is obstructed in V if H1 (S, D) , 0.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
S-maximal family
Let WC,S denote the unique irreducible component of HF V passing
through the point (C, S).
Definition 3.11
We call the image WC,S of WC,S in Hilb sc V is called the S-maximal family
containing C.
pr1
−→ Hilb sc V
∪
WC,S
↠
(C, S)
WC,S
∈
∈
HF V
∪
7−→
C
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Non-reduced components of Hilb sc V
For S-maximal families WC,S containing C, we give a sufficient condition
to be a generically smooth (resp. generically non-reduced) component of
Hilb sc V .
Corollary 3.12
In (1),(2),(3) of Main Theorem 2, we have furthermore that
1
2
3
If h1 (S, D) ≤ 1, then WC,S is an irreducible component of
(Hilb sc V)red .
Hilb sc V is generically smooth along WC,S if h1 (S, D) = 0, and
generically non-reduced along WC,S if h1 (S, D) = 1.
If H0 (S, −D) = 0, then dim[C] Hilb sc V = S · (KV )2 /2 + g + 1, where
g is the genus of C.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Examples
In the following examples, the closure W is a generically non-reduced
component Hilb sc V .
Example 3.13
Let V be a smooth quartic 3-fold V4 ⊂ P4 , E ⊂ V a smooth conic on V
with N E/V ≃ O2 , S ⊂ V a smooth hyperplane section of V containing E
and such that Pic S = Zh ⊕ ZE, where h ∼ OS (1).
Then a general member C of the complete linear system
Λ1 := |2h + 2E| ≃ | − 2KV S + 2E|
is a smooth connected curve of degree 12 and genus 13. Such curves C
sc
V4 of dimension
are parametrised by a loc. cl. irred. subset W ⊂ Hilb12,13
16.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Example 3.14
Let V = P3 and let F be a smooth plane cubic (elliptic) curve, S a smooth
quartic surface containing F. Then a general member C of
Λ2 := |4h + 2F| ≃ | − KV S + 2F|
is a smooth connected curve of degree 22 and genus 57. Such curves C
sc
are parametrised by a loc. cl. irred. subset W ⊂ Hilb22,57
P3 of dimension
90.
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Outline of Proof of Main Theorem 2(3)
Suppose that D := C + KV S ∼ (m + 1)F for an elliptic curve F on S and
m ≥ 1. Then there exists an exact sequence
p1
0 −→ A1 (C, S) −→ H0 (C, NC/V ) −→ H1 (S, −D) ≃ k m −→ 0.
We take α ∈ H0 (NC/V ) whose exterior component πC/S (α) is nonzero in
H1 (−D) ≃ k m but zero in H1 (−D + F) ≃ k m−1 . Then there exists an
infinitesimal deformation with pole β of S by the diagram
α
πC/S
−→
πC/S (α)
H0 (NS/V C ) ⊂
↓
H1 (− D)
→
≃ km
∈
∈
H0 (NC/V )
7−→
H0 (NS/V (F)) ∋ ∃ β
↓
H0 (NS/V (F)C )
↓
H1 (−D + F)
≃ k m−1
Now we apply the obstructedness theorem and finally we get
obS (α) , 0.
□
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§1 Introduction (Motivation)
§2 Infinitesimal analysis of the Hilbert scheme
§3 Obstructions to deforming curves lying on a K3 surface
Reference
D. Mumford.
Further pathologies in algebraic geometry.
Amer. J. Math., 84:642–648, 1962.
S. Mukai and H. Nasu,
Obstructions to deforming curves on a 3-fold I: A generalization of Mumford’s example
and an application to Hom schemes.
J. Algebraic Geom., 18(2009), 691-709
H. Nasu,
Obstructions to deforming curves on a 3-fold, II: Deformations of degenerate curves
on a del Pezzo 3-fold,
Annales de L’Institut Fourier, 60(2010), no.4, 1289-1316.
H. Nasu,
Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3
surface, preprint, arXiv:1601.07301.
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