Obstructions to deforming curves on a 3-fold, III: Deformations of curves lying on a K3 surface

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by  Hirokazu Nasu

OBSTRUCTIONS TO DEFORMING CURVES ON A 3-FOLD, III:
DEFORMATIONS OF CURVES LYING ON A K3 SURFACE
HIROKAZU NASU
6
1 Abstract. We study the deformations of a smooth curve C on a smooth projective 3-
0 foldV,assumingthepresenceofasmoothsurfaceS satisfyingC ⊂S ⊂V. Generalizing
2
aresultofMukaiandNasu,wegiveanewsufficientconditionforafirstorderinfinitesimal

n
deformation of C in V to be primarily obstructed. In particular, when V is Fano and
a
J S is K3, we give a sufficient condition for C to be (un)obstructed in V, in terms of

7 (−2)-curves and elliptic curves on S. Applying this result, we prove that the Hilbert
2
schemeHilbscV4ofsmoothconnectedcurvesonasmoothquartic3-foldV4 ⊂P4contains

] infinitely many generically non-reduced irreducible components, which are variations of
G
Mumford’s example for HilbscP3.
A
.
h
t
a
m 1. Introduction
[
Let V be a smooth projective 3-fold over an algebraically closed field k. This paper is a

1
v sequel to the preceding papers [17, 20], in which the embedded deformations of a smooth
1
curve C in V have been studied under the presence of an intermediate smooth surface S
0
3 satisfying C ⊂ S ⊂ V. As is well known, first order deformations C˜ ⊂ V ×Speck[t]/(t2)
7
0 of C in V are in one-to-one correspondence with global sections α of the normal bundle
1. N of C in V. Then the obstruction ob(α) to lifting C˜ to a second order deformation
C/V
60 C˜˜ ⊂ V ×Speck[t]/(t3) of C in V is contained in H1(C,NC/V), and ob(α) is computed as
1
a cup product α∪α by the map
:
v
Xi H0(C,NC/V)×H0(C,NC/V) −∪→ H1(C,NC/V)
r
a (cf. Theorem 2.1). It is generally difficult to compute ob(α) directly. Mukai and Nasu
[17] introduced the exterior components of α and ob(α), which are defined as the images
of α and ob(α) in Hi(C,N ) (i = 0,1) by the natural projection π : N →
S/V C C/S C/V
N , anddenotedbyπ (α(cid:12))andob (α), respectively (cf.§2.3). Theygave asufficient
S/V C C/S (cid:12) S
condi(cid:12)(cid:12)tion for obS(α) to be nonzero, which implies the non-liftability of C˜ to any C˜˜.
In this paper, we generalize their result and give a weaker condition for ob (α) 6= 0.
S
Let C˜ or α ∈ H0(C,N ) be a first order deformation of C in V, and π (α) ∈
C/V C/S
This work was partially supported by JSPS Grant-in-Aid (C), No 25400048, and JSPS Grant-in-Aid
(S), No 25220701.
2010 Mathematics Subject Classification. Primary 14C05; Secondary 14H10, 14D15.
Key words and phrases. Hilbert scheme, infinitesimal deformation, obstruction, K3 surface, Fano
threefold.
1
2 HIROKAZUNASU
H0(C,N ) the exterior component of α. Suppose that the image of π (α) in
S/V C C/S
H0(C,N ((cid:12)mE) ) lifts to a section
S/V(cid:12) C
(cid:12)
(cid:12) β ∈ H0(S,N (mE))
S/V
for an integer m ≥ 1 and an effective Cartier divisor E on S. In other words, we have
π (α) = β inH0(C,N (mE) ). Hereβ iscalledaninfinitesimal deformations with
C/S C S/V C
a pole (along(cid:12) E) of S in V (cf. §2.(cid:12)4). The following is a generalization of [17, Theorem
(cid:12) (cid:12)
2.2], in which, it was assumed that m = 1, E is smooth and irreducible with negative self-
intersection number E2 < 0 on S, and furthermore, E was assumed to be a (−1)-curve
on S (i.e. E ≃ P1 and E2 = −1) in its application.
Theorem 1.1. Let C˜, α, β be as above. Suppose that the natural map H1(O (kE)) →
S
H1(O ((k + 1)E)) is injective for every integer k ≥ 1. If the following conditions are
S
satisfied, then the exterior component ob (α) of ob(α) is nonzero:
S
(a) the restriction map H0(S,∆) −|→E H0(E,∆ ) is surjective for ∆ := C +K −
E V S
2mE, a divisor on S, and (cid:12) (cid:12)
(cid:12) (cid:12)
(b) we have
m∂ (β )∪β 6= 0 in H1(E,N ((2m+1)E −C) ),
E E E S/V E
(cid:12) (cid:12) (cid:12)
where β ∈(cid:12)H0(E,(cid:12)N (mE) ) is the principal part of β along(cid:12)the pole E, and
E S/V E
∂ is the(cid:12) coboundary map of th(cid:12)e exact sequence
E (cid:12) (cid:12)
π
(1.1) [0 −→ N −→ N −E→/S N −→ 0]⊗O (mE).
E/S E/V S/V E E
(cid:12)
≃OE(E) (cid:12)
|{z}
The relation between α and β is explained with Figure 1 in §3.
E
(cid:12)
Given a projective scheme V,(cid:12)let HilbscV denote the Hilbert scheme of smooth con-
nected curves in V. Mumford [18] first proved that HilbscP3 contains a generically non-
reduced (irreducible) component. Later, many examples of such non-reduced components
of HilbscP3 were found in [10, 3, 7, 5, 19], etc. More recently, Mumford’s example was
generalized in [17] and it was proved that for many uniruled 3-folds V, HilbscV contains
infinitely many generically non-reduced components. (See [23] for a different generaliza-
tion.) In the construction of the components, (−1)-curves E ⊂ V on a surface S ⊂ V
play a very important role. In this paper, as an application we study the deformations of
curves C on a smooth Fano 3-fold V when C is contained in a smooth K3 surface S ⊂ V.
On a K3 surface, (−2)-curves E (i.e. E ≃ P1 and E2 = −2) and elliptic curves F (then
F2 = 0) play a role very similar to that of a (−1)-curve. In what follows, we assume that
char(k) = 0.
Theorem 1.2. Let V be a smooth Fano 3-fold, S ⊂ V a smooth K3 surface, and C ⊂ S a
smooth connected curve, and put D := C+K a divisor on S. Suppose that there exists
V S
(cid:12)
(cid:12)
OBSTRUCTIONS TO DEFORMING CURVES ON A 3-FOLD, III 3
a first order deformation S˜ of S which does not contain any first order deformations C˜
of C.
(1) If D ≥ 0 and there exist no (−2)-curves and no elliptic curves on S, or more
generally, if H1(S,D) = 0, then HilbscV is nonsingular at [C].
(2) If D ≥ 0, 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,
(1.2) π (E) : H0(E,N (E)) −→ H0(E,N (E) )
E/S E/V S/V E
(cid:12)
is not surjective, then HilbscV is singular at [C]. (cid:12)
(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) is not surjective, then HilbscV is
F/S
singular at [C].
Thus C is unobstructed in V if H1(S,D) = 0. By using Theorem 1.1, we partially
prove that C is obstructed in V if H1(S,D) 6= 0. Under the assumption of Theorem 1.2,
the Hilbert-flag scheme HFV of V (cf. §2.5) is nonsingular at (C,S) (cf. Lemma 2.9).
Therefore (C,S) belongs to a unique irreducible component W of HFV. The image
C,S
W of W inHilbscV is called the S-maximal familycontaining C (cf. Definition 2.10).
C,S C,S
Then W is of codimension h1(S,D) in the tangent space H0(C,N ) of HilbscV at
C,S C/V
[C] (cf. (4.1)).
Corollary 1.3. In (1),(2),(3) of Theorem 1.2, we have furthermore that
(a) If h1(S,D) ≤ 1, then W is an irreducible component of (HilbscV) .
C,S red
(b) HilbscV is generically smooth along W if h1(S,D) = 0, and generically non-
C,S
reduced along W if h1(S,D) = 1.
C,S
(c) If H0(S,−D) = 0, then dim HilbscV = S·(K )2/2+g+1, where g is the genus
[C] V
of C.
The following is a simplification or a variation of Mumford’s example (See Examples
5.8 and 5.12 for more examples.).
Example 1.4. In the following examples, the closure W is an irreducible component of
(HilbscV) and HilbscV is generically non-reduced along W. (We have h0(C,N ) =
red C/V
dimW +1 at the generic point [C] of W).
(1) Let V be a smooth quartic 3-fold V ⊂ P4, E a smooth conic on V with trivial
4
normal bundle N ≃ O2, S a smooth hyperplane section of V containing E
E/V E
and such that PicS = Zh⊕ZE, where h ∼ O (1). Then a general member C of
S
the complete linear system |2h+2E| on S is a smooth connected curve of degree
12 and genus 13. Such curves C are parametrised by a locally closed irreducible
subset W of HilbscV of dimension 16.
4 HIROKAZUNASU
(2) 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 |4h + 2F| (h ∼ O (1)) is
S
a smooth connected curve of degree 22 and genus 57 on S. Such curves C are
parametrised by a locally closed irreducible subset W of HilbscP3 of dimension 90.
The organization of this paper is as follows. The proof of Theorem 1.1 heavily depends
on the analysis of the singularity of polar d-maps. Given a projective scheme V and its
hypersurface S ⊂ V, there exists a so-called “Hilbert-Picard” morphism ψ : HilbcdV →
S
PicS from the Hilbert scheme of effective Cartier divisors on V to the Picard scheme of
S, sending a hypersurface S′ ⊂ V to an invertible sheaf O (S′) on S. The tangent map
V S
d : H0(S,N ) → H1(S,O ) of ψ at [S] is called d-map for(cid:12)S ⊂ V. In §2.4, we show
S S/V S S (cid:12)
that this map is extended into a version dS◦ : H0(S,NS/V(mE)) → H1(S,OS((m+1)E))
with a pole along a divisor E ≥ 0 on S, and prove that for any β ∈ H0(S,N (mE))
S/V
the restriction dS◦(β) E to E of the image dS◦(β) coincides with the coboundary ∂E(β E)
of (1.1) up to the mu(cid:12)ltiple m (cf. Proposition 2.5). In §3, applying this result to a(cid:12)3-
(cid:12) (cid:12)
fold V, we prove Theorem 1.1. In § 4, we prove Theorem 1.2 and Corollary 1.3 as an
application of Theorem 1.1. In §5.1, we study the Mori cone NE(S) of a smooth quartic
surface S of Picard number two (cf. Lemmas 5.1, 5.3). Applying the results on Mori
cones, in §5.2, for curves C lying on S, we study the deformations of C in P3 and C in
a smooth quartic 3-fold V ⊂ P4. In particular, we give a sufficient condition for W to
4 C,S
be a generically non-reduced (or generically smooth) component of HilbscP3 and HilbscV
4
(cf. Theorems 5.5, 5.10 and 5.13).
Acknowledgments. I should like to thank Eiichi Sato for asking me a useful question
concerning the existence of a non-reduced component of the Hilbert scheme of a smooth
Fano 3-fold of index one. I am grateful to Jan Oddvar Kleppe for helpful comments and
a discussion on non-reduced components of the Hilbert scheme of space curves.
2. Preliminaries
We start by recalling some basic facts on the deformation theory of closed subschemes,
as well as setting up Notations. We work over an algebraically closed field k of charac-
teristic p ≥ 0. Let V ⊂ Pn be a closed subscheme of Pn with the embedding invertible
sheaf O (1) on V, and X ⊂ V a closed subscheme of V with the Hilbert polynomial
V
P = χ(O (n)). Then as is well known, the Hilbert scheme Hilb V of V parametrises
X X P
all closed subscheme X′ of V with PX′ = PX (cf. [6]). We denote by HilbV the (full)
Hilbert scheme Hilb V of V. Let I and N = (I /I2)∨ denote the ideal sheaf
P P X X/V X X
and the normalFsheaf of X in V, respectively. The symbol [X] represents the point of
HilbV corresponding to X. Then the tangent space of HilbV at [X] is known to be
isomorphic to the group Hom(I ,O ) of sheaf homomorphisms from I to O , which is
X X X X
isomorphic to H0(X,N ). Every obstruction to deforming X in V is contained in the
X/V
OBSTRUCTIONS TO DEFORMING CURVES ON A 3-FOLD, III 5
group Ext1(I ,O ), and if X is a locally complete intersection in V, then it is contained
X X
in a smaller subgroup H1(X,N ) ⊂ Ext1(I ,O ). If H1(X,N ) = 0, then HilbV
X/V X X X/V
is nonsingular at [X] of dimension h0(X,N ). A first order (infinitesimal) deformation
X/V
of X in V is a closed subscheme X′ ⊂ X × SpecD, flat over the ring D = k[t]/(t2) of
dual numbers, with a central fiber X′ = X. By the universal property of the Hilbert
0
scheme, there exists a one-to-one correspondence between the set of D-valued points
γ : SpecD → HilbV sending 0 to [X], and the set of the first order deformations of X in
V. By the infinitesimal lifting property of smoothness (cf. [8, Proposition 4.4, Chap. 1]),
if there exists a first order deformation of X in V not liftable to a deformation over
Speck[t]/(tn) for some n ≥ 3, then HilbV is singular at [X]. We say X is unobstructed
(resp. obstructed) in V if HilbV is nonsingular (resp. singular) at [X], and for an irre-
ducible closed subset W of HilbV, we say HilbV is generically smooth (resp. generically
non-reduced) along W if HilbV is nonsingular (resp. singular) at the generic point X of
η
W.
2.1. Primary obstructions. Let V be a (projective) scheme over k, X a closed sub-
scheme of V, α a global section of N . We define a cup product ob(α) ∈ Ext1(I ,O )
X/V X X
by
ob(α) := α∪e∪α,
where e ∈ Ext1(O ,I ) is the extension class of the standard short exact sequence
X X
(2.1) 0 −→ I −→ O −→ O −→ 0
X V X
on V. Though the following fact is well-known to the experts, we give a proof for the
reader’s convenience.
Theorem 2.1 (cf. [2, 13]). Let X˜ be a first order deformation of X in V corresponding
to α. If X is a locally complete intersection in V, then X˜ lifts to a deformation X˜˜ over
k[t]/(t3), if and only if ob(α) is zero.
Proof. First we recall the correspondence between X˜ and α. Let U := U i ∈ I be an
i
open affine covering of V, R = Γ(U ,O ) the coordinate ring of U , I(cid:8) =(cid:12)Γ(U ,I(cid:9) ) the
i i V i i (cid:12) i X
defining ideal ofX∩U inU . Wetakethecovering Uso thattheintersection U := U ∩U
i i ij i j
is affine for all i,j and let R := Γ(U ,O ). Let m be the codimension of X in V and
ij ij V
let I be generated by elements f ,...,f of R . Since the two ideals I and I agree on
i i1 im i i j
the overlap U , there exist a m×m matrix A with entries in R (i.e., A ∈ M(m,R ))
ij ij ij ij ij
such that
f
i1
.
(2.2) fj = Aijfi, where fi :=  .. .
f 
 im
6 HIROKAZUNASU
The restriction A to X of A represents the transition matrix of N . Throughout the
ij ij X/V
proof, for a section u of a sheaf F onV, udenote the imageof u inF = F⊗O . Let I
X X X˜
betheidealsheafdefiningX˜ inV×Speck[t]/(t2). TheidealJ := Γ(U(cid:12) ×Speck[t]/(t2),I )
i (cid:12)i X˜
of R ⊗k[t]/(t2)(≃ R ⊕R t) is generated by
i i i
f +tu , ..., f +tu
i1 i1 im im
for some u ∈ R (k = 1,...,m). Since J and J agree on U × Speck[t]/(t2), there
ik i i j ij
exists a matrix B ∈ M(m,R ) such that
ij ij
u
i1
.
fj +tuj = (Aij +tBij)(fi +tui), where ui :=  .. .
u 
 im
Comparing the coefficient of t, we have
(2.3) u = A u +B f ,
j ij i ij i
which implies that
u = A u .
j ij i
Let α be the section of N ≃ Hom(I ,O ) over U sending each f to u (k =
i X/V X X i ik ik
1,...,m), respectively. Then by (2.2) and (2.3), the local sections α over U agree on
i i
the overlap U and define a global section α ∈ Hom(I ,O ). Conversely, every global
ij X X
section α of N defines a first order deformation X˜ of X in V, whose local equations
X/V
are f + tα\(f ) (k = 1,...,m), where α\(f ) is a lift of α (f ) ∈ R /I to R . In the
ik i ik i ik i ik i i i
rest of the proof, for convenience, we write as α (f ) = u instead of writing α (f ) = u
i i i i ik ik
(k = 1,...,m).
Now we try to lift the first order deformation X˜ to a second order deformation X˜˜ of
X in V over k[t]/(t3). Suppose that X˜ lifts to a deformation X˜˜ over k[t]/(t3). Then the
ideal K := Γ(U ×Speck[t]/(t3),I ) of R ⊗k[t]/(t3)(≃ R ⊕R t⊕R t2) is generated by
i i X˜˜ i i i i
f +tu +t2v , ..., f +tu +t2v
i1 i1 i1 im im im
for some v ∈ R (k = 1,...,m). Since K and K agree on U × Speck[t]/(t3), there
ik i i j ij
exists a matrix C ∈ M(m,R ) such that
ij ij
v
i1
.
fj +tuj +t2vj = (Aij +tBij +t2Cij)(fi +tui +t2vi), where vi :=  .. .
v 
 im
Then we have v = A v +B u +C f , and in particular, we have
j ij i ij i ij i
(2.4) v −A v = B u .
j ij i ij i
X˜˜ is defined as a subscheme of V × k[t]/(t3), flat over D = k[t]/(t3) if and only if we
can solve the equation (2.4) for v . Let us define a section β of N over U by
i ij X/V ij
OBSTRUCTIONS TO DEFORMING CURVES ON A 3-FOLD, III 7
β (f ) = v −A v . Then since A is the transition matrix of N , we note that the
ij i j ij i ij X/V
1-cochain β := {β } of N becomes a coboundary (of a 0-cochain β′ = {β′} of N
ij X/V i X/V
with β′(f ) = v ). Let us define a section γ of N over U by
i i i ij X/V ij
(2.5) γ (f ) = B u .
ij i ij i
Then the 1-cocycle {γ } ∈ C1(U,N ) is cohomologous to zero if and only if we can
ij X/V
solve (2.4) for v . Thus it suffices to prove that {γ } represents the class of the cup
i ij
product ob(α) = α∪e∪α in H1(N ) ⊂ Ext1(I ,O ).
X/V X X
Let δ : Hom(I ,O ) → Ext1(I ,I ) be the map induced by (2.1). Applying the
X X X X
functor Hom(I ,∗) to (2.1), we obtain an exact sequence of Cˇech complexes
X
0 −→ C•(U,Hom(I ,I )) −→ C•(U,Hom(I ,O )) −→ C•(U,Hom(I ,O )) −→ 0.
X X X V X X
We compute the image δ(α)(= α ∪ e) of α by a diagram chase. Let α := α ∈
i Ui
Γ(U ,Hom(I ,O )) as before and let α (f ) = u for each i. Since U is affine, (cid:12)there
i X X i i i i (cid:12)
exists a section α˜ of Hom(I ,O ) over U (a local lift of α over U ) satisfying α˜ (f ) = u .
i X V i i i i i
Since α is globally defined, δ(α) = α˜ −α˜ becomes a section of Hom(I ,I ) over U
ij j i X X ij
for every i,j. Then by (2.3), we have δ(α) (f ) = B f . Thus we have computed the
ij i ij i
image δ(α) in H1(Hom(I ,I )) ⊂ Ext1(I ,I ). Since the cup product of δ(α) with α
X X X X
is locally defined as the composition of the two maps δ(α) and α , it is represented by a
ij i
1-cocycle {α ◦δ(α) } of Hom(I ,O ). Therefore δ(α)∪α = α∪e∪α is contained in
i ij X X
H1(N ) ⊂ Ext1(I ,O ). Since we have
X/V X X
α ◦δ(α) (f ) = α (B f ) = B α (f ) = B u ,
i ij i i ij i ij i i ij i
we conclude that α ◦δ(α) = γ by (2.5). Thus we have completed the proof. (cid:3)
i ij ij
Definition 2.2. Here ob(α) is called the (primary) obstruction for α (or X˜).
2.2. Hypersurface case and d-map. Let X be an effective Cartier divisor on V, i.e.,
a closed subscheme of V whose ideal sheaf is locally generated by a single equation. We
denote by HilbcdV the Hilbert scheme of effective Cartier divisors on V. There exists
a natural morphism ϕ : HilbcdV −→ PicV to the Picard scheme PicV of V, sending a
divisor D to the invertible sheaf O (D). We define a morphism ψ : HilbcdV → PicX
V X
by the composition of ϕ with the restriction map O (D) 7−→ O (D) to X. By definition,
V X
the tangent map d of ψ at [X] is the composite
X X
(2.6) d : H0(X,N ) −→δ H1(V,O ) −|X→ H1(X,O ),
X X/V V X
where δ is the coboundary map of the exact sequence 0 → O → O (X) → N → 0.
V V X/V
We call d the d-map for X. Let X˜ be a first order deformation of X in V, correspond-
X
ing to a global section β of N . Then by [20, Lemma 2.9], the primary obstruction
X/V
ob(β) of X˜ equals the cup product d (β)∪β by the map H1(X,O )×H0(X,N ) →∪
X X X/V
H1(X,N ).
X/V
8 HIROKAZUNASU
2.3. Exterior components. We recall the definition of the exterior components (cf. [17,
20]), which is useful for computing the obstructions to deforming subschemes of codi-
mension greater than 1. Let X,Y be two closed subschemes of V such that X ⊂ Y,
π : N → N the natural projection. Then we have the induced maps
X/Y X/V Y/V X
Hi(π ) : Hi(N ) →(cid:12) Hi(N ) on the cohomology groups for i = 0,1. The two
X/Y X/V (cid:12) Y/V X
images (cid:12)
(cid:12)
π (α) := H0(π )(α) and ob (α) := H0(π )(ob(α))
X/Y X/Y Y X/Y
are called the exterior component of α and ob(α), respectively. These objects respectively
correspond to the deformation of X in V into the normal direction to Y and its obstruc-
tion. Now we assume that Y is an effective Cartier divisor on V, X is a locally complete
intersection on Y. The d-map d in (2.6) is generalized and defined for a pair (X,Y) (We
X
have d = d ).
Y Y,Y
Definition 2.3. Let δ be the coboundary map of [0 → I → O → O → 0]⊗O (Y).
X V X V
The composition
d : H0(X,N ) −→δ H1(V,O (Y)⊗I ) −|→X H1(X,N ⊗N ∨)
X,Y Y/V X V X Y/V X X/V
(cid:12) (cid:12)
(cid:12) (cid:12)
of δ with the restriction map | to X is called the d-map for (X,Y).
X
Then the two d-maps d and d are related by the following commutative diagram:
X,Y Y
H0(Y,N ) −−d−Y→ H1(Y,O )
Y/V Y
|X

(2.7) |X H1(Xy,OX)

y H1(ι)
H0(X,N ) −d−X−,→Y H1(X,N ∨ ⊗N ),
Y/V X X/yV Y/V
(cid:12)
where ι : O → N ∨ ⊗N i(cid:12)s the sheaf homomorphism induced by π .
X X/V Y/V X/Y
Lemma 2.4 (cf. [17, Lemma 2.3 and 2.4]). Let X˜ and Y˜ be first order deformations of
X and Y in V, with the corresponding global sections α,β of N ,N , respectively. If
X/V Y/V
we have π (α) = β in H0(N ), then we have
X/Y X Y/V X
(cid:12) (cid:12)
(cid:12) (cid:12)
(2.8) ob (α) = d (π (α))∪ α = d (β) ∪ π (α)
Y X,Y X/Y 1 Y X 2 X/Y
(cid:12)
in H1(N ), where ∪ and ∪ are the cup product ma(cid:12)ps
Y/V X 1 2
(cid:12)
(cid:12) H1(X,N ∨ ⊗N )×H0(X,N ) −∪→1 H1(X,N ),
X/V Y/V X X/V Y/V X
(cid:12) (cid:12)
H1(X,O (cid:12))×H0(X,N ) −∪→2 H1(X,N (cid:12) )
X Y/V X Y/V X
(cid:12) (cid:12)
(cid:12) (cid:12)
respectively.
OBSTRUCTIONS TO DEFORMING CURVES ON A 3-FOLD, III 9
2.4. Infinitesimal deformations with poles and polar d-maps. In this section, we
recall the theory of infinitesimal deformations with poles, which was introduced in [17].
Here we develop the study [17, §2.4] on the polar d-maps further. The infinitesimal
deformations with poles are defined as rational sections of some sheaves admitting a pole
along some divisor, and they are usually regarded as the deformations of the open objects
complementary to the poles.
Let V be a projective scheme, Y and E effective Cartier divisors on V and Y, respec-
tively. Put Y◦ := Y \ E and V◦ := V \ E and let ι : Y◦ ֒→ Y be the open immersion.
Since ι∗OY◦ contains OY(mE) as a subsheaf for any m ≥ 0, there exists natural in-
clusions OY ⊂ OY(E) ⊂ ··· ⊂ OY(mE) ⊂ ··· ⊂ ι∗OY◦ of sheaves on Y. Similarly,
since NY/V(mE) ⊂ ι∗NY◦/V◦ for any m ≥ 0, we regard H0(NY/V(mE)) as a subgroup of
H0(NY◦/V◦) by the natural injective map
H0(Y,NY/V(mE)) ֒→ H0(Y◦,NY◦/V◦).
A rational section β of N admitting a pole along E, i.e.
Y/V
β ∈ H0(Y,N (mE))
Y/V
for m ≥ 1 is called an infinitesimal deformation of Y with a pole along E. Every infinites-
imal deformation with a pole induces a first order deformation of Y◦ in V◦ by the above
injection.
Now we assume that the natural map
(2.9) H1(Y,O (mE)) −→ H1(Y,O ((m+1)E))
Y Y
is injective for any m ≥ 1. Then by the same argument as in [17, Lemma 2.5], the natural
map
H1(Y,OY(mE)) −→ H1(Y◦,OY◦)
is injective for any m ≥ 1. By this map, we regard H1(O (mE)) as a subgroup of
Y
H1(OY◦) for any m ≥ 1. Given an invertible sheaf L on Y, we identify an element of
H1(O (mE)) as a first order deformation of the invertible sheaf L◦ := ι L on Y◦, and
Y ∗
call it an infinitesimal deformation of L with a pole along E.
Let β be a global section of N (mE) for m ≥ 1, i.e., an infinitesimal deformation
Y/V
of Y with a pole along E. Let dY◦ : H0(NY◦/V◦) → H1(OY◦) be the d-map (2.6) for
Y◦ ⊂ V◦. We have a generalization of [17, Proposition 2.6], which enables us to compute
the singularity of dY◦(β) ∈ H1(OY◦) along the boundary E.
Proposition 2.5. Let m ≥ 1 be an integer.
(1) dY◦(H0(Y,NY/V(mE))) ⊂ H1(Y,OY((m+1)E)).
10 HIROKAZUNASU
(2) Let dY be the restriction of dY◦ to H0(Y,NY/V(mE)), and let ∂E be the coboundary
map of (1.1). Then the diagram
H0(Y,N (mE)) −−d−Y→ H1(Y,O ((m+1)E))
Y/V Y
|E |E
 
H0(E,N  (mE) ) −−m−∂→E H1(E,O ((m+1)E))
Yy/V E Ey
(cid:12)
is commutative. (cid:12)
In other words, if Y is a hypersurface in V, then every infinitesimal deformation of
Y ⊂ V with a pole induces that of its invertible sheaf NY/V. The principal part of dY◦(β)
along E coincides with the coboundary ∂ (β ) of the principal part β , up to constant.
E E E
(cid:12) (cid:12)
Proof. Let U := {U } be an affine open co(cid:12)vering of V and let x = y(cid:12) = 0 be the local
i i∈I i i
equation of E over U such that y defines Y in U . Through the proof, for a local section
i i i
t of a sheaf F on V, t¯denotes the restriction t ∈ F for conventions. Let D and D
Y Y xi x¯i
denote the affine open subsets of U and U ∩Y(cid:12)defined(cid:12) by x 6= 0 and x¯ 6= 0, respectively.
i i (cid:12) (cid:12) i i
Then {D } is an affine open covering of Y◦ since D = D ∩Y = U ∩Y◦.
x¯i i∈I x¯i xi i
Let β be a global section of N (mE) ≃ O (Y)(mE). Then the product x¯mβ is
Y/V Y i
contained in H0(U ,O (Y)) and lifts to a section s′ ∈ Γ(U ,O (Y)) since U is affine. In
i Y i i V i
particular, β lifts to the section si := s′i/xmi of OV◦(Y◦) over Dxi. Let δ : H0(NY◦/V◦) →
H1(OV◦) be thecoboundary mapin thedefinition (2.6) for a hypersurface Y◦ ⊂ V◦. Then
we have
s′ s′
δ(β)ij = sj −si = xmj − xmi in Γ(Dxi ∩Dxj,OV◦(Y◦))
j i
foreveryi,j. Sinceβ isaglobalsectionofNY◦/V◦, δ(β)ij iscontainedinΓ(Dxi∩Dxj,OV◦).
Put f := xmxmδ(β) = xms′ −xms′. Since xms = s′ ∈ Γ(U ,O (Y)) for every i, f is
ij i j ij i j j i i i i i V ij
a section of O . Now we recall the relations between the local equations x ,y of E over
Uij i i
U . Since the two ideals (x ,y ) and (x ,y ) agree on the overlaps U , there exist elements
i i i j j ij
b and c of O satisfying x = b y +c x . Then we have
ij ij Uij i ij j ij j
f = (xm −cmxm)s′ +xm(cms′ −s′)
ij i ij j j j ij j i
m−1
= xm−1−k(c x )k(x −c x )s′ +xm(cms′ −s′)
i ij j i ij j j j ij j i
X
k=0
m−1
(2.10) = xm−1−k(c x )kb y s′ +xm(cms′ −s′).
i ij j ij j j j ij j i
X
k=0
Since y ∈ Γ(U ,O (−Y)) and s′ ∈ Γ(U ,O (Y)), b y s′ is a section of O , while
j j V j j V ij j j Uij
cms′ −s′ ∈ Γ(U ,O (Y)) is also a section of O , because we have
ij j i ij V Uij
cms′ −s′ = c¯mx¯mβ −x¯mβ = (c¯mx¯m −x¯m)β = 0
ij j i ij j i ij j i