Seshadri constants and degrees of defining polynomials

0.16 MB·English

by  Atsushi Ito

SESHADRI CONSTANTS AND DEGREES OF DEFINING
POLYNOMIALS
3
1 ATSUSHIITO ANDMAKOTO MIURA
0
2 Dedicated to Professor Yujiro Kawamata on the occasion of his sixtieth birthday.

n
a
Abstract. In this paper, we study a relation between Seshadri con-
J
stants and degrees of defining polynomials. In particular, we compute
1
the Seshadriconstants on Fano varieties obtained as complete intersec-
3
tions in rational homogeneous spaces of Picard numberone.

]
G
A
. 1. Introduction
h
t
a Seshadri constant was introduced by Demailly in [Dem], as an invariant
m which measures the local positivity of ample line bundles.
[
Definition 1.1. Let L be an ample line bundle on a projective variety X,

1 and take a (possibly singular) closed point p ∈ X. The Seshadri constant of
v
L at p is defined to be
3
3 ε(X,L;p) := max{t ≥ 0|µ∗L−tE is nef},
6
7
where µ : X → X is the blowing up at p and E = µ−1(p) is the exceptional
.
1 divisor.
0
Equivaleently, the Seshadri constant can be also defined as
3
1
C.L
: ε(X,L;p) = inf ,
iv C (cid:26)multp(C)(cid:27)
X
where the infimum is taken over all reduced and irreducible curves C on
r X passing through p. We call a curve C a Seshadri curve of L at p if
a
ε(X,L;p) = C.L/mult (C).
p
Seshadri constants have many interesting properties (see [La, Chapter
5] for instance), but it is very difficult to compute them in general. Many
authorsstudysurfacecases, butcomputationsinhigherdimensionsarerare.
Let X be a projective variety embedded in PN and p ∈ X. The pur-
pose of this paper is to study the Seshadri constant ε(X,O (1);p) by in-
X
vestigating homogeneous polynomials which define X. It is easy to see
that ε(X,O (1);p) ≥ 1 for such X and p. Furthermore it is known that
X
2010 Mathematics Subject Classification. 14C20.
Key words and phrases. Seshadriconstant,definingpolynomial,rationalhomogeneous
space.
1
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 2
ε(X,O (1);p) = 1 holds if and only if there exists a line on X passing
X
through p (cf. [Ch, Lemma 2.2]).
In [Ba] and [Ch], Bauer and Chan give a lower bound of ε(X,O (1);p)
X
by using the degree deg(X) := O (1)dimX when X is a surface and a 3-fold
X
respectively.
Theorem1.2(cf.[Ba,Theorem2.1],[Ch,Theorem1.4]). LetX beasmooth
projective surface or a 3-fold in PN, and p ∈ X a point. If there exists no
line on X passing through p, it holds that
deg(X)
ε(X,O (1);p) ≥ .
X
deg(X)−1
Remark 1.3. Bauer proved the sharpness of the lower bound as well, that
is, for each d ≥ 3, there exist a smooth surface X and p ∈ X such that
d= deg(X) and ε(X,O (1);p) = d/(d−1). Chan also constructed such X
X
and p in 3-dimensional case for d ≥ 4, although X might have finitely many
singular points.
Insteadofthedegree, weintroduced (X)foraprojectivevariety X ⊂ PN
p
and p ∈ X (cf. [La, Definition 1.8.37]).
Definition 1.4. Let X be a projective variety in PN and p ∈ X a point.
We define d (X) to be the least integer d such that the natural map
p
H0(PN,I ⊗O (d))⊗O → I ⊗O (d)
X PN PN X PN
is surjective at p, where IX ⊂ OPN is the ideal sheaf corresponding to X. In
other words, X is cut out scheme theoretically by hypersurfaces of degree
d (X) at p.
p
By using d (X), we give a lower bound of the Seshadri constant on X ⊂
p
PN (which may be singular) in any dimensions.
Theorem 1.5. Let X be a projective variety in PN and p ∈ X a point. If
there exists no line on X passing through p, it holds that
d (X)
p
ε(X,O (1);p) ≥ .
X
d (X)−1
p
Furthermore, this lower bound is sharp, i.e., for any n ≥ 1 and d ≥ 2,
there exist a smooth projective variety X ⊂ PN and p ∈ X such that n =
dimX, d = d (X), and ε(X,O (1);p) = d/(d−1).
p X
Remark 1.6. It holds d (X) ≤ deg(X) for any projective variety X ⊂ PN
p
and p ∈X (see the proof of [Mu, Theorem 1]). Thus Theorem 1.5 improves
Theorem 1.2 even in the cases dimX =2,3.
In Section 3, we compute the Seshadri constants on some varieties X ⊂
PN by finding a curve C such that deg(C)/mult (C) coincides with the
p
lower bound d (X)/(d (X)−1). As a special case, we obtain the following
p p
theorem.
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 3
Theorem 1.7 (=Corollary 3.4). Let Y ⊂ PN be a rational homogeneous
space of Picard number 1, which is embedded by the ample generator. Let
X be a complete intersection variety in Y of hypersurfaces of degrees d ≤
1
... ≤ d such that −K = O (1) and p ∈ X. If there exists no line on X
r X X
passing through p, it holds that
d (X) d /(d −1) when d ≥ 2,
ε(X,O (1);p) = p = r r r
X dp(X)−1 (cid:26) 2 when dr = 1.
Remark 1.8. If X is a complete intersection variety in Y of hypersurfaces of
degrees d ≤ ... ≤ d such that −K = O (i) for i ≥ 2, it is easy to check
1 r X X
that X is covered by lines (see Lemma 2.2). Thus ε(X,O (1);p) = 1 holds
X
for any p ∈ X.
Hence Theorem 1.7 states that we can compute ε(X,O (1);p) for any
X
Fano variety X obtained as a complete intersection in any rational homoge-
neous space of Picard number 1.
Throughout this paper, all schemes are defined over the complex number
field C. In Section 2, we give the proof of Theorem 1.5. In Section 3, we
compute Seshadri constants on some varieties.
Acknowledgments. The authors would like to express their gratitude to
Professor Yujiro Kawamata for his valuable advice, comments, and warm
encouragement. They are also grateful to Professors Katsuhisa Furukawa
and Kiwamu Watanabe for their useful comments and suggestions.
2. Lower bounds
In this paper, a line means a projective curve of degree 1 in PN. The
moduli of lines plays an important role in this paper.
Definition 2.1. Let X be a projective scheme in PN and p ∈ X a point.
WedenotebyF (X)themodulispaceoflinesonX passingthroughp. Note
p
that F (X) is naturally embedded in F (PN)∼= PN−1.
p p
For a graded ring S, we denote by S the set of all homogeneous elements
i
of degree i. We will use the following lemma in Sections 2, 3.
Lemma 2.2. Let X be a projective scheme in PN and p ∈ X a point. Fix
homogeneous coordinates x ,...,x on PN such that p = [1 : 0 : ··· : 0].
0 N
Assume that X is defined by homogeneous polynomials {f } around p,
j 1≤j≤r
and write f = dj xdj−ifi for d = degf and fi ∈ C[x ,...,x ] . Then
j i=1 0 j j j j 1 N i
F (X) ⊂ F (PN) ∼= ProjC[x ,...,x ] is defined by {fi} .
p p P 1 N j 1≤j≤r,1≤i≤dj
Proof. Theproofissimilartothatof Proposition2.13 (a)in[Deb]. Weleave
the details to the reader. (cid:3)
For a variety X, a point p ∈ X, and an effective Cartier divisor D, we
define ord (D) := max{m ∈ N|f ∈ mm}, where f is a defining function of
p p
D at p. The following lemma is easy and well known, but we prove it for
the convenience of the reader.
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 4
Lemma 2.3. Let C be a curve on a projective variety X and p ∈ C a point.
For an effective Cartier divisor D on X not containing C, it holds
C.D ≥ ord (D)·mult (C).
p p
Proof. Let ν : C → C be the normalization. Then there exists an effective
divisor E on C such that OCe(−E) = ν−1mC,p and deg(E) = multp(C).
Since O (−D|e)⊂ mordp(D|C), we have
C C C,p
e
OCe(−ν∗D|C)⊂ OCe(−ordp(D|C)E).
This means
D.C = deg(ν∗D| )≥ deg(ord (D| )E)
C p C
= ord (D| )·mult (C).
p C p
Since ord (D| )≥ ord (D), this lemma is proved. (cid:3)
p C p
Now we can prove Theorem 1.5. The idea is simple. For any curve C on
X passing through p, we find a suitable divisor D ∈|O (i)| not containing
X
C for some i and apply Lemma 2.3.
Proof of Theorem 1.5. Let x ,...,x be homogeneous coordinates on
0 N
PN such that p = [1 : 0 : ··· : 0], and set d = d (X). Choose and fix a basis
p
f ,...,f of H0(PN,I ⊗O (d)) ⊂ C[x ,...,x ] . As in Lemma 2.2, we
1 r X PN 0 N d
can write
f = xd−1f1+xd−2f2+···+x fd−1+fd
j 0 j 0 j 0 j j
for some fi ∈ C[x ,...,x ] .
j 1 N i
For 1 ≤ i ≤ d−1, 1 ≤ j ≤ r, we set
Di = (xi−1f1+···+x fi−1+fi = 0) ⊂ PN.
j 0 j 0 j j
Note that Di can be PN itself. By the definition of Di, it holds
j j
X ∩ Di ⊂ (f = 0) ∩ Di
j j j
1≤i≤d−1,1≤j≤r 1≤j≤r 1≤i≤d−1,1≤j≤r
(∗) \ \ \
= (fi = 0).
j
1≤i≤d,1≤j≤r
\
By the definition of d (X), we have X = (f = ··· = f = 0) around p.
p 1 r
Hence the last term of (∗) is nothing but ConeF (X) by Lemma 2.2, where
p
ConeF (X)istheprojectiveconeofF (X)inPN withthevertexatp. Since
p p
F (X) = ∅ by assumption, we have
p
(†) Di| = {p}.
j X
1≤i≤d−1,1≤j≤r
\
Fix a curve C ⊂ X passing through p. To show the inequality in this
theorem, it suffices to show
deg(C) d
≥ .
mult (C) d−1
p
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 5
By (†), there exist 1 ≤ i≤ d−1, 1 ≤j ≤ r such that Di| does not contain
j X
C. In particular, Di| is an effective divisor on X not containing C. Since
j X
xd−i(xi−1f1+···+x fi−1+fi) = −(xd−i−1fi+1···+x fd−1+fd) on X,
0 0 j 0 j j 0 j 0 j j
it holds that
xi−1f1+···+x fi−1+fi
ord (Di| )= ord 0 j 0 j j
p j X p xi
0 (cid:12)X!
(cid:12)
xd−i−1fi+1···+x fd−1+(cid:12)fd
0 j 0 j (cid:12) j
= ord − ≥ i+1.
p xd0 (cid:12)X!
(cid:12)
Since Di| ∼ O (i), we have (cid:12)
j X X (cid:12)
i deg(C)= (Di| ).C ≥ ord (Di| )·mult (C) ≥ (i+1)mult (C)
j X p j X p p
by Lemma 2.3. Hence it holds deg(C)/mult (C)≥ (i+1)/i ≥ d/(d−1) by
p
1≤ i ≤ d−1, and the inequality of this theorem is shown.
Now, we show the sharpness of the lower bound. First, assume d≥ n+1.
Set
f = xd−1f1+···+xd−n+1fn−1+x fd−1+fd ∈ C[x ,...,x ]
0 0 0 0 n+1
for a general fi ∈ C[x ,...,x ] for each 1 ≤ i ≤ n−1 and i = d−1,d.
1 n+1 i
Then f defines a smooth hypersurface X ⊂ Pn+1 containing p = [1 : 0: ··· :
0]. By the generality of fi,
F (X) = (f1 = ··· = fn−1 = fd−1 = fd = 0) ⊂ ProjC[x ,...,x ]
p 1 n+1
is empty. Set
C = (f1 = ··· = fn−1 = x fd−1+fd = 0) ⊂ Pn+1.
0
By definition, C is contained in X and contains p. Since all fi are general,
C is a complete intersection curve. Hence
deg(C)= (n−1)!·d, mult (C)= (n−1)!·(d−1)
p
hold, and we have
ε(X,O (1);p) ≤ deg(C)/mult (C) = d/(d−1).
X p
Since d (X) = d and F (X) = ∅, it holds ε(X,O (1);p) = d/(d−1) by the
p p X
inequality of this theorem.
When d ≤ n, we use Theorem 1.7, which is proved in the next section.
(Of course, we do not use the sharpness assertion in Theorem 1.5 to show
Theorem 1.7.) For 2 ≤ d ≤ n, choose positive integers r and d ≤ ··· ≤ d
1 r
such that d= d and r d = n+r.
r j=1 j
Applying Theorem 1.7 to Y = Pn+r, we have ε(X,O (1);p) = d /(d −
P X r r
1) = d/(d − 1) for a smooth complete intersection X of hypersurfaces of
degrees d ≤ ... ≤ d and general p ∈ X. Note that we can easily check
1 r
F (X) = ∅ for general p ∈ X by using Lemma 2.2. Since d (X) = d = d,
p p r
the sharpness is proved. (cid:3)
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 6
3. Finding Seshadri curves
In [It], the first author computes Seshadri constants on some Fano mani-
folds at a very general point. In the paper, toric degenerations are used to
estimate Seshadri constants from below. Since the lower semicontinuity is
used there for lower bounds, we have to assume some very generality in the
method. Instead of toric degenerations, we use the lower bound in Theorem
1.5 here. For upper bounds, we find Seshadri curves similar to [It]. That is,
for some variety X, we can find a curve C ⊂ X passing through p such that
deg(C)/mult (C) coincides with the lower bound in Theorem 1.5.
p
To show Theorem 1.7, we treat the case d ≥ 2 in Subsection 3.1. In
r
that case, we construct a Seshadri curve by cutting a suitable cone in X by
hypersurfaces. In Subsection 3.2, we treat the case d = 1. In that case,
r
we show the existence of a conic C ⊂ X passing through p by using the
deformation theory. In Subsection 3.3, we prove Theorem 1.7.
3.1. Cutting cones by hypersurfaces. In this subsection, we prove the
following theorem, from which the case d ≥ 2 in Theorem 1.7 follows im-
r
mediately.
Theorem 3.1. Let Y be a projective variety in PN and p ∈ Y a point. For
a subvariety X ⊂ Y containing p, we assume the following:
i) Around p, X is a locally complete intersection in Y of hypersurfaces
of degrees d ≤ ... ≤ d . That is, r = codim(X,Y) and there exists
1 r
f ∈H0(PN,O (d )) for each j such that
j PN j
X = Y ∩ (f = 0)
j
1≤j≤r
\
holds in a neighborhood of p.
ii) F (Y)6= ∅ and r d ≤ dimF (Y)+1.
p j=1 j p
iii) d (Y)≤ d .
p r
P
Then it holds that
1 when F (X) 6= ∅,
ε(X,O (1);p) = p
X d /(d −1) when F (X) = ∅.
r r p
(cid:26)
Proof. When F (X) 6= ∅, this theorem is clear. Thus we may assume
p
F (X) = ∅, and show ε(X,O (1);p) = d /(d − 1). As in the proof of
p X r r
Theorem 1.5, we take homogeneous coordinates x ,...,x on PN such that
0 N
p = [1 : 0 : ··· : 0], and write f = dj xdj−ifi for fi ∈ C[x ,...,x ] . By
j i=1 0 j j 1 N i
Lemma 2.2, we have
P
F (X) = F (Y) ∩ (fi = 0) ⊂ ProjC[x ,··· ,x ].
p p j 1 N
i,j
\
Hence F (X) is an intersection of F (Y) and d hypersurfaces. Thus
p p j
r d = dimF (Y)+1 must hold by the condition ii) and F (X) = ∅.
j=1 j p P p
P
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 7
By Theorem 1.5, we have ε(X,O (1);p) ≥ d /(d − 1) since d (X) ≤
X r r p
max{d (Y),d }= d . To show the opposite inequality, we may assume that
p r r
each fi is general because of the lower semicontinuity of Seshadri constants
j
(cf. [La, Example 5.1.11]). For general fi, we can find a curve C ⊂ X
j
through p such that deg(C)/mult (C)= d /(d −1) as follows.
p r r
Fix an irreducible component Z of F (Y) such that dimZ = dimF (Y).
p p
We define a curve C ⊂ PN to be
C = ConeZ ∩ (f1 = ··· = fdj−1 = fdj = 0)
j j j
1≤j≤r−1
\
∩(f1 = ··· = fdr−2 = x fdr−1+fdr = 0).
r r 0 r r
Note d ≥ 2 holds because X is not linear and d (X) ≤d . Since
r p r
ConeZ ⊂ ConeF (Y) ⊂ Y,
p
C is contained in X. By definition, C is cut out from ConeZ by r−1d +
i=1 j
(d − 1) = r d − 1 hypersurfaces, and r d = dimF (Y) + 1 =
r j=1 j j=1 j pP
dimConeZ. Since all fi are general, C is a reduced and irreducible curve.
P j P
By definition, we have
deg(C) = deg(ConeZ)·d !···d !·(d −2)!·d ,
1 r−1 r r
mult (C) = mult (ConeZ)·d !···d !·(d −2)!·(d −1).
p p 1 r−1 r r
Thus it holds that deg(C)/mult (C) = d /(d −1) because deg(ConeZ) =
p r r
deg(Z) = mult (ConeZ). Hence ε(X,O (1);p) ≤ deg(C)/mult (C) =
p X p
d /(d −1) holds and this theorem follows. (cid:3)
r r
3.2. Finding conics. When Y ⊂ PN is a rational homogeneous space of
Picard number 1 other than a projective space, d (Y) = 2 for any p ∈ Y.
p
Hence we cannot apply Theorem 3.1 to the case when d = 1, i.e., X is a
r
section of Y by hyperplanes.
Since d (X) = d (Y) = 2 holds for such X, the lower bound obtained
p p
by Theorem 1.5 is d (X)/(d (X)−1) = 2. Thus if there exists a (smooth)
p p
conic C ⊂ X passing through p, we have
deg(C)
2≤ ε(X,O (1);p) ≤ = 2.
X
mult (C)
p
Forthefollowingproposition,wepreparesomenotations. Forasubvariety
X in a variety Y, we denote by I the ideal sheaf on Y corresponding to
X/Y
X. A conic in PN is a smooth projective curve of degree 2, and a plane in
PN is a 2-dimensional linear projective subspace. For a projective variety
Y ⊂ PN,wesay thatY is covered by lines(resp.conics, planes)ifforgeneral
p ∈Y, there exists a line (resp. a conic, a plane) on Y containing p.
Proposition 3.2. Let Y ⊂ PN be a smooth projective variety satisfying the
following:
i) IY/PN ⊗OPN(2) is globally generated,
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 8
ii) for a general p ∈ Y, dimR (Y) = 2 holds, where R (Y) is the
p p
subscheme of the Hilbert scheme Hilb(Y) which parametrizes conics
on Y passing through p.
Then for general p ∈ Y and general hyperplane H ⊂ PN containing p,
there exists a conic C on Y ∩H passing through p.
Proof. Fix a general point p ∈ Y. Since dimR (Y) = 2, we can choose an
p
irreducible component R of R (Y) of dimension 2. We define the incidence
p
variety I as
I = {(C,H) ∈ R×|OPN(1)⊗mp||C ⊂ H},
and consider the natural projections
π(cid:0)(cid:0)(cid:1)(cid:1)1(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)I LLLLLLLLπL2LLL&&
R |OPN(1)⊗mp|.
Toshow this proposition, wehave toshow thattheprojection π is gener-
2
ically surjective. For a fixed C ∈ R, a hyperplane H ∈ |O (1)⊗m | con-
PN p
tains C if and only if H contains the plane spanned by C. Thus π is a
1
PN−3-bundle and it holds that
dimI = dimR+N −3 = N −1 = dim|O (1)⊗m |.
PN p
Hence it suffices to show that dimπ−1(H) = 0 for a general H ∈ π (I).
2 2
Step 1. In this step, we show the following claim.
Claim 3.3. In the above setting, the composition of the natural maps
H0(PN,I ⊗O (1))⊗O → H0(Y,I ⊗O (1))⊗O
C/PN PN PN C/Y Y Y
→ I ⊗O (1)
C/Y Y
is surjective for C ∈R.
Proof of Claim 3.3. Fix C ∈ R, and let P ⊂ PN be the plane spanned by
C
C. Since dimR (Y) = 2 and dimR (P ) = 4, P is not contained in Y.
p p C C
Choose a general section f ∈ H0(PN,IY/PN ⊗ OPN(2)), and let D ⊂ PN
be the corresponding hypersurface of degree 2. By the condition i), the
generality of f, and P 6⊂ Y, we have P 6⊂ D . Thus as an effective divisor
C C
on P , C is contained in D| . Since C is a conic and D| is an effective
C PC PC
divisor of degree 2 on P ∼= P2, C and D| coincide as schemes. This
C PC
means IC/PN = ID/PN +IPC/PN. Thus it holds that
I = I /I = (I +I )/I
C/Y C/PN Y/PN D/PN PC/PN Y/PN
= (IY/PN +IPC/PN)/IY/PN.
The last equality follows from I ⊂ I . Since
D/PN Y/PN
H0(PN,IPC/PN ⊗OPN(1))⊗OPN → IPC/PN ⊗OPN(1)
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 9
is surjective and H0(PN,I ⊗O (1)) = H0(PN,I ⊗O (1)), this
PC/PN PN C/PN PN
claim follows. (cid:3)
Step 2. Let (C,H) ∈ I be a general element, and set X = Y ∩H. In this
step, we show that X is smooth. (Note that we do not know H is general in
|OPN(1)⊗mp| a priori. Thus we have to check the smoothness of X.) It is
clear that X\C is smooth by Claim 3.3 and the generality of (C,H). Hence
it is enough to show that X is smooth along C.
Consider
B := {(q,H) ∈ C ×|O (1)⊗I ||Y ∩H is singular at q}.
PN C/PN
For each q ∈ C, the fiber of the projection B → C over q corresponds to the
kernel of
H0(PN,IC/PN ⊗OPN(1)) → H0(Y,IC/Y ⊗OY(1))
→ (I +m2 )/m2 ,
C/Y Y,q Y,q
where m is the maximal ideal sheaf on Y at q. By Claim 3.3, this
Y,q
map is surjective. Hence the projection B → C is a Pk-bundle for k =
dim|OPN(1)⊗IC/PN|−codim(C,Y). Thus we have
dimB = k+1 = dim|O (1)⊗I |+2−dimY.
PN C/PN
IfdimY ≥ 3, thenaturalprojection B → |OPN(1)⊗IC/PN|isnotgenerically
surjective since dimB < dim|O (1)⊗I |. This means X = Y ∩H is
PN C/PN
smooth along C for general H ∈ |O (1) ⊗ I |. When dimY = 2, it
PN C/PN
holds K .C = −4 because dimR = 2 and C is a free rational curve. Thus
Y
we have C2 =2. By the Hodge index theorem, Y ⊂ PN is a quadric surface
and C ∼ O (1). Hence X must coincide with C, which is smooth.
Y
Step 3. Let (C,H) ∈ I and X = Y ∩ H be as in Step 2. To prove
dimπ−1(H) = 0, it is enough to show N ∼= O⊕n−1 for n = dimX
2 C/X C
because dimπ−1(H) ≤ h0(C,N ⊗m ). Write
2 C/X p
n
f∗NC/Y = OP1(ai)
i=1
M
for integers a ≥ ... ≥ a , where f is an isomorphism P1 → C. (We use f
1 n
not to confuse O (1)| and the degree 1 invertible sheaf on C.) Since C is
PN C
free on Y, it follows that a ≥ 0 for any i. Furthermore, a = dimR= 2
i i i
holds since C is free. Hence we have
P
f∗NC/Y = OP1(2)⊕O⊕n−1 or OP1(1)⊕2 ⊕O⊕n−2.
Let α ∈ H0(PN,I ⊗O (1)) be a section corresponding to H. From
C/PN PN
the natural surjection
I ⊗O (1) → I /I2 ⊗O (1)| ∼= N∨ ⊗O (1)|
C/Y Y C/Y C/Y Y C C/Y Y C
SESHADRI CONSTANTS AND DEGREES OF DEFINING POLYNOMIALS 10
and Claim 3.3, we obtain a surjective map
δ :H0(PN,I ⊗O (1))⊗O → N∨ ⊗O (1)| .
C/PN PN C C/Y Y C
Furthermore, there exist natural isomorphisms
N∨ ⊗O (1)| ∼= Hom (N ,O (1)| )∼= Hom (N ,N | ).
C/Y Y C C C/Y Y C C C/Y X/Y C
The image δ(α) ∈ H0(C,N∨ ⊗O (1)| ) ∼= Hom (N ,N | ) of α
C/Y Y C C C/Y X/Y C
induces an exact sequence
δ(α)
0 → N → N → N | → 0.
C/X C/Y X/Y C
Since f∗NC/Y = OP1(2)⊕O⊕n−1 or OP1(1)⊕2 ⊕O⊕n−2 and f∗NX/Y|C =
OP1(2), we have NC/X ∼= OC⊕n−1 if the restriction of δ(α) on O(2) or O(1)⊕2
is surjective. Since α is general, this follows from the subjectivity of δ. (cid:3)
3.3. Proof of Theorem 1.7. As a corollary of Theorem 3.1 and Proposi-
tion 3.2, we obtain Theorem 1.7. As stated in Remark 1.8, Theorem 1.7 can
be rephrased as follows.
Corollary 3.4 (=Theorem 1.7). Let Y ⊂ PN be a rational homogeneous
space of Picard number 1, which is embedded by the ample generator. Let
X be a complete intersection variety in Y of hypersurfaces of degrees d ≤
1
... ≤ d such that −K is ample. For p ∈ X, it holds that
r X
1 when F (X) 6= ∅,
p
ε(X,O (1);p) = d /(d −1) when F (X) = ∅ and d ≥ 2,
X r r p r

2 when F (X) = ∅ and d = 1.
 p r
Proof. By the adjunction formula, −K is ample if and only if r d ≤
 X j=1 j
i(Y) − 1, where i(Y) is the positive integer satisfying −K = O (i(Y)),
Y Y
P
i.e., the Fano index of Y. It is well known that dimF (Y) = i(Y)−2. For
p
instance, the existence of lines is proved in [Ko, Theorem V.1.1.15], and
dimF (Y)is computed by the deformation theory. By Lemma 2.2, it is easy
p
to show that X is covered by lines if r d < i(Y) − 1 as in the first
j=1 j
paragraph of the proof of Theorem 3.1.
Furthermore, Y ⊂ PN is cut out byPquadrics, i.e., d (Y) ≤ 2 holds for
p
any p ∈ Y (see [Li] for example). Thus we can apply Theorem 3.1 if d ≥ 2,
r
and this corollary follows in that case.
Assume F (X) = ∅ and d = 1. In this case, i(Y) = r d +1 = r+1
p r j=1 j
andY is not aprojective space. Sinced (X) ≤ d (Y) =2, ε(X,O (1);p) ≥
p p X
P
2 follows from Theorem 1.5.
To show the opposite inequality, we use Proposition 3.2. By the lower
semicontinuity of Seshadri constants, we may assume X = Y ∩ r H for
j=1 j
general hyperplanes H ⊂ PN and p ∈ X is a general point. Set
j T
r−1
Y′ = Y ∩ H ,
j
j=1
\