Rational surfaces in index-one Fano hypersurfaces

0.18 MB·English

by  Roya Beheshti

RATIONAL SURFACES IN INDEX-ONE FANO
HYPERSURFACES
6
0
0
2 ROYABEHESHTIANDJASONMICHAELSTARR

n
a Abstract. We give the firstevidence for aconjecture that ageneral, index-
J one,Fanohypersurfaceisnotunirational: (i)ageneralpointofthehypersur-

9 face is contained inno rational surfaceruled, roughly, by low-degree rational
2 curves,and(ii)ageneralpointiscontainedinnoimageofaDelPezzosurface.

]
G
A
1. Introduction
.
h
t For complex, projective varieties a classical notion is unirationality: A variety
a
rationallydominatedbyprojectivespaceisunirational. Amodernnotionisrational
m
connectedness: Avariety is rationally connected if everypairof points is contained
[
inarationalcurve. Everyunirationalvarietyisrationallyconnected. The2notions

1 agree for curves and surfaces. Conjecturally they disagree in higher dimensions.
v
0 Conjecture 1.1. For every integer n ≥ 4 there exists a non-unirational, smooth,
1 degree-n hypersurface in Pn.
7
1 A smooth hypersurface in Pn of degree d ≤ n is an index-(n +1− d), Fano
0 manifold. By [2] and [9], every Fano manifold is rationally connected. Versions of
6
Conjecture 1.1 have been aroundfor decades. The specific case n=4 is attributed
0
to Iskovskikh and Manin, [7].
/
h In [8], Koll´ar suggested an approachto proving Conjecture 1.1. A generalpoint
t
a of an n-dimensional, unirational variety is contained in a k-dimensional, rational
m subvariety for each k < n. Thus, Conjecture 1.1 for n ≥ 5 follows from the next
: conjecture (which fails for n=4).
v
i Conjecture 1.2. For every integer n ≥5, there exists a smooth, degree-n hyper-
X
surface in Pn whose general point is contained in no rational surface.
r
a
We give the first evidence for Conjecture 1.2.
Theorem 1.3. For every integer n≥5, every smooth, degree-n hypersurface X in
Pn contains a countable union of closed, codimension-2 subvarieties containing the
image of every generically-finite, rational transformation P1×P1 99KX mapping a
general fiber {t}×P1 isomorphically to an (n−1)-normal, smooth, rational curve
in X.
Theorem 1.4. For every integer n ≥ 5, every smooth, degree-n hypersurface in
Pn contains a countable union of closed, codimension-2 subvarieties containing the
image of every generically finite, regular morphism from a Del Pezzo surface to X.
Date:February2,2008.
1
Aprojectivevarietyisk-normal ifeveryglobalsectionoftherestrictionofOPn(k)
is the restriction of a global section on Pn.
Thereare2approaches. First,givenarationalsurfaceS andaregularmorphism
f :S →X,to provedeformationsoff arecontainedina codimension-2subvariety
of X, it suffices to prove n−4(f∗T /T )/Torsion has only the zero section. In
X S
V
Section 2, this is used to prove Theorems 1.3 and 1.4.
Second,arationalsurfacewithapencilofrationalcurvesgivesamorphismfrom
P1 to a parameterspaceof rationalcurvesonX. There is a canonicalconstruction
of algebraic differential forms on the parameter space. Since P1 has only the zero
form, these forms limit rational curves on the parameter space. In Section 3, this
is used to prove the following generalizationof Theorem 1.3.
Theorem 1.5. For every integer n≥5, every smooth, degree-n hypersurface X in
Pn contains a countable union of closed, codimension-2 subvarieties containing the
image of every generically-finite, rational transformation S 99K X from a surface
with a pencil of curves mapping the general curve isomorphically to an (n−1)-
normal, smooth curve of genus 0 or 1, also assumed non-degenerate if the genus is
1.
As the second approach does not apply to Theorem 1.4, the first approach is
more productive. However, further progress in proving Conjecture 1.2 will likely
use both approaches,as well as new ideas.
Acknowledgments. The authors thank Mike Roth for many illuminating discus-
sions.
2. The first approach
Let X be a smooth,degree-nhypersurfacein Pn, n≥5. Denote by Hilb(X)the
Hilbert scheme of X. Theorem 1.3 follows easily from the next theorem.
Theorem 2.1. Let Z be an irreducible subscheme of Hom(P1,Hilb(X)) satisfying,
(i) the associated morphism Z ×P1 → Hilb(X) does not factor through the
projection Z×P1 →Z, and
(ii) theimageofageneralpointofZ×P1parametrizesasmooth,(n−1)-normal,
rational curve in X.
Thenthereexistsacodimension≥2subvarietyofX containingallcurvesparametrized
by Z×P1.
A morphism P1 → Hilb(X) is equivalent to a closed subscheme S′ ⊂ P1 ×X,
flat over P1. If a general point of P1 parametrizes a smooth rational curve, then
S′ is an irreducible surface. Any desingularization S of S′ is a surface fitting in a
diagram,
f
S X (1)
π
P1
2
Proposition 2.2. Let Z be an irreducible subvariety of Hom(P1,Hilb(X)) satisfy-
ing,
(i) the associated morphism Z ×P1 → Hilb(X) does not factor through the
projection Z×P1 →Z,
(ii) a general point of Z ×P1 parametrizes a smooth curve in X, and
(iii) thereisnocodimension2subvarietyofX containingallcurvesparametrized
by Z×P1.
Then, for the morphism P1 → Hilb(X) parametrized by a general point of Z, the
torsion-free sheaf n−4(f∗T /T )/Torsion has a nonzero global section.
X S
V
Proof. Replacing Z by a dense, Zariski open subset if necessary, assume Z is
smooth. Let V ⊂Z×P1×X be the pullback of the universal family to Z×P1
Let g : V → V be a desingularization of V. Denote by φ the projection map
fromV to Ze×X,and denote by p the projectionmap fromV to Z. Let φ=φ◦g,
and let p˜=p◦g. e
g˜
g φ
V V Z×X
e
Z×P1 p
p˜
Z.
Replacing Z by a dense, Zariski open subset if necessary, assume p˜ is smooth,
cf. [6, Corollary III.10.7]. Associated to the morphism g˜ is the derivative map,
dg˜:T →g˜∗T .
V X
Associated to the morphism p˜is the deerivative map,
dp˜:T →p˜∗T .
V Z
By hypothesis, dp˜is surjective. Denoteeby T the kernel of dp˜. Because Z×P1 →
p˜
Hilb(X)does notfactorthroughZ, the restrictionofg˜to ageneralfiber ofp˜maps
genericallyfinitelytoitsimage. Thereforethefollowingmapisgenericallyinjective,
dg˜:T →g˜∗T .
p˜ X
As T is locally free and V is integral, in fact dg˜ is injective. Denoting by N the
p˜
cokernel of g˜∗T by dg˜(Te), there is a commutative diagram with exact rows
X p˜
0 Tp˜ TV dp˜ p˜∗TZ 0
e
= dg˜ u
0 Tp˜ g˜∗TX N 0
Bygenericsmoothness,the rankofdg˜atageneralpointofV equalsthe dimension
of the closure of Image(g˜). By hypothesis, this is ≥n−2. Teherefore the rank of u
at a general point is ≥n−4. Thus, the restriction of u to a general (n−4)-plane
in the fiber of p˜∗T has rank n−4. A general (n−4)-plane is the tangent space
Z
3
of a general (n−4)-dimensional subvariety of Z. Therefore, after replacing Z by
the smooth locus of a general (n−4)-dimensional subvariety of Z, assume Z is
(n−4)-dimensional and u is generically injective.
Associated to u, there is an induced map,
n−4 n−4 n−4
(u):p˜∗ T → N/Torsion.
^ ^ Z ^
Because u is generically injective and n≥5, this map is generically injective.
Let z be a generalpoint of Z, and denote by S and S the fibers of p and p˜over
z, respectively. Since V is smooth, S is a smooth surfaece. Let f : S → X be the
restriction of φ to S, aend let f˜: S →e X be the corresponding map from S. The
restrictionofN toS ispreciselyf∗eT /T . Therestrictionofp˜∗T toS ispreecisely
X S Z
the trivial vectorbundle TZ,z⊗COS. Since z is general,the restrictionof n−4(u)
V
is generically injective. Therefore, it is a nonzero map,
n−4 n−4 n−4
^(u)|S :(^ TZ,z)⊗COS →(^ f∗TX/TS)/Torsion.
Since T is (n−4)-dimensional, this is equivalent to a nonzero global section of
Z,z
( n−4f∗T /T )/Torsion (well-defined up to nonzero scaling). (cid:3)
X S
V
Proposition 2.3. Let P1 → Hilb(X) be a morphism with associated diagram as
in Equation 1. If the curve parametrized by a general point of P1 is smooth and
(n−1)-normal then
n−2
h0(S,ω ⊗ f∗T )=0.
S ^ X
Proof. Pulling back the short exact sequence of tangent bundles
0→TX →TPn|X →NX/Pn ∼=OX(n)→0
to S and taking its (n−1)st exterior power gives another short exact sequence
n−1
n−1 n−2
0→ ^ f∗TX →^ f∗TPn →f∗OX(n)⊗^ f∗TX →0.
Tensoring this sequence with ω ⊗f∗O (−n) gives the following short exact se-
S X
quence
n−1 n−1 n−2
0→ωS⊗f∗OX(−n)⊗^ f∗TX →ωS⊗f∗OX(−n)⊗^ f∗TPn →ωS⊗^ f∗TX →0.
Applying the long exactsequence of cohomology,h0(S,ω ⊗ n−2f∗T ) equals
S X
V
0 if both,
(i) h0(S,ωS ⊗f∗OX(−n)⊗ n−1f∗TPn) equals 0, and
(ii) h1(S,ω ⊗f∗O (−n)⊗Vn−1f∗T ) equals 0.
S X X
V
Proof of (i). Consider the Euler exact sequence on Pn
0→OPn →OPn(1)n+1 →TPn →0.
Pulling this back to S, and taking its nth exterior power gives the following exact
sequence
n−1
0→^ f∗TPn →f∗OX(n)⊕(n+1) →f∗OX(n+1)→0.
4
Tensoring with ω ⊗f∗O (−n) gives an injective map
S X
ωS ⊗f∗OX(−n)⊗^n−1f∗TPn →ωS⊕(n+1).
Thusit suffices to proveh0(S,ω ) equals0,whichfollowsfromthe hypothesisthat
S
S is a rational surface.
Proof of (ii). There is a canonical isomorphism
ω ⊗f∗O (−n)⊗ n−1f∗T ∼=ω ⊗f∗O (−n+1).
S X ^ X S X
So by Serre duality, it suffices to prove h1(S,f∗O (n−1)) equals 0. Let C be a
X
general fiber of the map π :S →P1. There is a short exact sequence
0→f∗O (n−1)⊗I →f∗O (n−1)→f∗O (n−1)| →0, (2)
X C/S X X C
where I is the ideal sheaf of C in S. By hypothesis, the image of C by f is
C/S
(n−1)-normal in Pn, therefore the map
H0(S,f∗O (n−1))→H0(C,f∗O (n−1)| )
X X C
is surjective. The long exact sequence of cohomology to the sequence in Equation
2 gives an isomorphism
H1(S,f∗O (n−1)⊗I )∼=H1(S,f∗O (n−1)). (3)
X C/S X
Because S is a smooth surface and the general fiber of π is a smooth, rational
curve, R1π∗F is the zero sheaf for every π-relatively globally-generated, coher-
ent O -module F. Because f∗O (n−1) is globally-generated, it is π-relatively
S X
globally-generated. BecauseIC/S equalsπ∗OP1(−1), the twistf∗OX(n−1)⊗IC/S
is π-relatively globally-generated. Thus R1π∗(f∗OX(n−1)) and R1π∗(f∗OX(n−
1)⊗I ) are each the zero sheaf. So, by the Leray spectral sequence, there are
C/S
canonical isomorphisms
H1(S,f∗OX(n−1))∼=H1(P1,π∗(f∗OX(n−1))), (4)
H1(S,f∗OX(n−1)⊗IC/S)∼=H1(P1,π∗(f∗OX(n−1)⊗IC/S)). (5)
Taken together, Equations 3, 4 and 5 give a canonical isomorphism
H1(P1,π∗(f∗OX(n−1)))∼=H1(P1,π∗(f∗OX(n−1))⊗OP1(−1)).
This is possible only if h1(P1,π∗(f∗OX(n−1))) equals 0. (cid:3)
Proof of Theorem 2.1. Let Z satisfy the hypotheses of Proposition 2.2, and let S
andf satisfy the conclusionofProposition2.2. The injective mapdf :T →f∗T
S X
induces a multiplication map,
2 n−4 n−2
T ⊗ f∗T → f∗T .
^ S ^ X ^ X
The image sheaf is precisely 2T ⊗( n−4(f∗T /T ))/Torsion. Tensoring with
X X S
V V
the canonical bundle of ω , this gives an injective map
S
n−4 n−2
( (f∗T /T ))/Torsion→ω ⊗ f∗T .
^ X S S ^ X
By hypothesis, ( n−4(f∗T /T ))/Torsionhas a nonzero global section. Therefore
X S
ω ⊗ n−2f∗T Valso has a nonzero global section.
S X
V
5
On the other hand, for Z satisfying the hypothesis of Theorem 2.1, Proposition
2.3 implies,
n−2
h0(S,ω ⊗ f∗T )=0.
S ^ X
Thus Z does not satisfy the hypothesis of Proposition 2.3, i.e., it does not satisfy
Hypothesis(iii). Thereforethereexistsacodimension2subvarietyofX containing
all the curves parametrized by Z×P1. (cid:3)
Proof of Theorem 1.3. For every generically-finite, rational transformation P1 ×
P1 99K X restricting to a closed immersion on a general fiber, there is an asso-
ciated rational transformation
P1 99KHilb(X), t7→Image({t}×P1).
By properness of the Hilbert scheme and the valuative criterion, this extends
to a regular morphism. Therefore, associated to each rational transformation
is an element in the Hom-scheme Hom(P1,Hilb(X)). Those rational transfor-
mations satisfying the hypothesis of Theorem 1.3 give a locally closed subset of
Hom(P1,Hilb(X)). As Hom(P1,Hilb(X)) is a countable union of quasi-projective
varieties, this subset is also a countable union of quasi-projective subvarieties. By
Theorem 2.1, for each such subvariety Z, there is a codimension-2 subvariety of X
containingeverycurveparametrizedbyZ×P1. Thissubvarietycontainstheimage
ofeachrationaltransformationP1×P1 99KX givingapointinZ. Therefore,there
exists a countable union of codimension-2 subvarieties of X containing the image
of every rational transformation satisfying the hypothesis of Theorem 2.1. (cid:3)
The proof of the Theorem1.4 is similar to the proof of Theorem 2.1. There is a
preliminary proposition.
Proposition 2.4. Let X be a smooth hypersurface of degree n in Pn. For every
Del Pezzo surface S and every generically finite morphism f : S → X, the only
global section of n−4(f∗T /T )/Torsion is the zero section.
X S
V
Proof. The proof is similar to the proof of Theorem 2.1. By the same type of
argument, it suffices to prove,
(i) h0(S,ωS ⊗f∗OX(−n)⊗ n−1f∗TPn) equals 0, and
(ii) h1(S,ω ⊗f∗O (−n)⊗Vn−1f∗T ) equals 0.
S X X
V
Theproofof(i)isthesameasintheproofofTheorem2.1,sinceh0(S,ω )equals
S
0. As for (ii), there is a canonical isomorphism
ω ⊗f∗O (−n)⊗ n−1f∗T ∼=(ω−1⊗f∗O (n−1))−1.
S X ^ X S X
Denote ω−1 ⊗f∗O (n−1) by L. The sheaf f∗O (n−1) is globally generated.
S X X
By the hypothesis that S is a Del Pezzo surface, ω−1 is ample. Thus L is ample.
S
By Kodaira vanishing, h1(S,L−1) equals 0. So, using the canonical isomorphism,
h1(S,ω ⊗f∗O (−n)⊗ n−1f∗T ) equals 0. (cid:3)
S X X
V
6
Proof of Theorem 1.4. By the same countability argument as at the beginning of
the section, it suffices to prove that for every flat family
φ
D X ×V
p
V
such that φ is generically finite and a general fiber of p is a Del Pezzo surface, the
image of φ is contained in a subvariety of codimension ≥ 2. Let S be the fiber
of p over a general point of V, and let f be the restriction φ| : S → X. As in
S
the proof of the Proposition 2.2, if the image of φ is contained in no subvariety
of codimension ≥ 2, then H0(S, n−4(f∗T /T )/Torsion) has a nonzero global
X S
V
section. Thus Proposition2.4 proves the image of φ is contained in a subvariety of
codimension ≥2. (cid:3)
3. The second approach
Let X be a smooth hypersurface in Pn of degree n. We denote by M (X) the
g,n
Kontsevich moduli stack of families of genus-g, n-pointed, stable maps to X. The
associated coarse moduli space is denoted by M (X), cf. [4].
g,n
ForeveryintegralsubschemeM ofM (X),denotebyX(M)thesmallestclosed
g,0
subvariety of X containing every curve parametrized by M.
Hypothesis 3.1. Let M be an integral, closed subscheme of M (X).
g,0
(i) The curves parametrized by M are contained in no codimension 2 sub-
scheme of X, i.e., dim(X(M))≥n−2.
(ii) The integer g equals 0 or 1.
(iii) A general point of M parametrizes an embedded, smooth, (n−1)-normal
curve. If g equals 1, also the curve is nondegenerate.
(iv) The dimension of M equals dim(X)−2=n−3.
Theorem 3.2. Forevery integern≥4,for every smooth, degree-n hypersurface X
in Pn, and for every integral closed subscheme M of M (X) satisfying Hypothesis
g,0
3.1, every desingularization of M has a nonzero canonical form. In particular, M
is not uniruled.
There are 2 components of the proof: a global construction of certain (n−3)-
forms on M following [3], and a local description of the forms proving they are
nonzero. The construction is 3.3, the local description is Lemma 3.4, and the
nonvanishing result is Claim 3.5.
Let M be a finite type scheme and let ν :M →M (X) be a morphism. Later,
g,0
M willfbe a desingularization of an integralfsubscheme M of M (X). For every
g,0
ifnteger p, [3, Corollary 4.3] gives a map,
ψ :H1(X,Ωp+1)→H0(M (X),Ωp ).
p X g,0 Mg,0(X)
Denote,
M := M × M (X) ,
stack (cid:16) Mg,0(X) g,0 (cid:17)red
f f
7
i.e., the associated reduced stack of the 2-fibered product. There are projections,
π :M →M,
1 stack
f f
π :M →M (X).
2 stack g,0
There are associated pullback mapfs on p-forms,
π∗ :H0(M,Ωp )→H0(M ,Ωp ),
1 M stack Mstack
f f
f f
π∗ :H0(M (X),Ωp )→H0(M ,Ωp ).
2 g,0 Mg,0(X) stack Mstack
f
Hence there is a map, f
π∗◦ψ :H1(X,Ωp+1)→H0(M ,Ωp ).
2 X stack Mstack
f
Does this map factor through π∗, i.e., does there exist afmap,
1
φ :H1(X,Ωp+1)→H0(M,Ωp ),
p X M
f
such that π∗◦φ equals π∗◦ψ ? f
1 p 2 p
Lemma 3.3. If M is smooth there is a unique map of C-vector spaces,
f φ :H1(X,Ωp+1)→H0(M,Ωp ),
p X M
f
such that π∗◦φ equals π∗◦ψ . f
1 p 2 p
Proof. Since M is smooth and M is reduced, [3, Proposition 3.6] implies the
stack
pullback map,f f
H0(M,Ωp )→H0(M ,Ωp ),
stack
M Mstack
is an isomorphism. f f f f (cid:3)
Let C be a smooth curve in X with corresponding point [C] in M (X). For
g,0
every integer p, restriction to the fiber at [C] defines a map,
ψ :H1(X,Ωp+1)→Ωp | .
p,[C] X Mg,0(X) [C]
TheZariskitangentspacetoM (X)at[C]isH0(C,N ). Thedualvectorspace
g,0 C/X
is the fiber of Ω at [C]. The pth exterior power is the fiber of Ωp at
Mg,0(X) Mg,0(X)
[C]. Therefore ψ is equivalent to a linear map,
p,[C]
p
ψ :H1(X,Ωp+1)→Hom( H0(C,N ),C).
p,[C] X ^ C/X
What is the map ψ ? In other words, for an element in H1(X,Ωp+1), what is
p,[C] X
the associated p-linear alternating map on H0(C,N )?
C/X
When X is a smooth hypersurface in Pn and p = n−3, the answer follows as
in [3, Theorem 5.1]. For a smooth hypersurface X in Pn, Griffiths computed the
cohomology groups Hq(X,Ωp ), cf. [5, Section 8]. In particular, there is an exact
X
sequence,
H0(X,Ωn−1(X)| )→H0(X,Ωn (2X)| )→H1(X,Ωn−2)→0. (6)
Pn X Pn X X
ThereforeeveryelementofH1(X,Ωn−2)istheimageβofanelementβinH0(X,Ωn (2X)| ).
X Pn X
There is a short exact sequence of locally free O -modules,
C
0→NC/X →NC/Pn →NX/Pn|C →0.
8
Taking the (n−2)nd exterior power gives a short exact sequence,
n−2 n−2 n−3
0→ ^ NC/X → ^ NC/Pn →(^ NC/X)⊗NX/Pn|C →0.
Twisting each term by N∨ | gives an exact sequence,
X/Pn C
n−2 n−2 n−3
0→ ^ NC/X ⊗NX∨/Pn|C → ^ NC/Pn ⊗NX∨/Pn|C → ^ NC/X →0. (7)
Applying the long exact sequence of cohomology,there is a connecting map,
n−3 n−2
H0(C, N )→H1(C, N ⊗N∨ | ).
^ C/X ^ C/X X/Pn C
Now n−2N is the determinant of N , which is canonically isomorphic
C/X C/X
to ωC⊗V(ΩXn−1)∨|C. By adjunction, ΩXn−1 is isomorphic to ΩnPn(X)|X. Also, NX/Pn
is isomorphic to OPn(X)|X. Putting this together gives a canonical isomorphism,
n−2
^ NC/X ⊗NX∨/Pn|C ∼=ωC ⊗(ΩnPn(2X)∨)|C.
Serre duality gives an isomorphism,
H1(C,ωC ⊗(ΩnPn(2X)∨)|C)∼=H0(C,ΩnPn(2X)|C)∨.
The pullback map H0(X,Ωn (2X)| ) → H0(C,Ωn (2X)| ) determines a trans-
Pn X Pn C
pose map,
H0(C,Ωn (2X)| )∨ →H0(X,Ωn (2X)| )∨.
Pn C Pn X
Finally, every element β of H0(X,Ωn (2X)| ) determines a linear functional,
Pn X
H0(X,Ωn (2X)| )∨ →C.
Pn X
Puttingallthistogether,everyelementβ ofH0(X,Ωn (2X)| )determinesalinear
Pn X
functional,
n−3
β :H0(C, N )→C.
^ C/X
e
Lemma 3.4. Let X be a smooth hypersurface in Pn. For every element β of
H0(X,Ωn (2X)| ), φ (β) equals the restriction of β to n−2H0(C,N ),
Pn X n−3,[C] C/X
V
up to nonzero scaling. e
9
Proof. This follows by a diagram-chase. Here are the main points. There is a
commutative diagram with exact rows and columns,
0 0 (8)
Id
T T
C C
0 TX|C TPn|C NX/Pn|C 0
Id
0 NC/X NC/Pn NX/Pn|C 0
0 0
The following three invertible sheaves are isomorphic,
n−2
ωC ⊗ ^ NC∨/X ∼=ΩXn−1|C ∼=ΩnPn(X)|C.
Denote any by L. Twisting Equation 8 by L gives a commutative diagram with
exact rows and columns,
0 0
0 Vn−2NC∨/X ∼= Vn−1NC∨/Pn ⊗OPn(X)|C 0
0 ΩnX−2|C ΩnPn−1(X)|C ΩnX−1|C⊗OPn(X)|C 0
∼=
0 ωC⊗Vn−3NC∨/X ωC⊗Vn−2NC∨/Pn ⊗OPn(X)|C ωC⊗Vn−2NC∨/X ⊗OPn(X)|C 0
0 0 0
(9)
For a sheaf E on X, E(X)|X denotes the tensor product E ⊗OPn(X)|X. And for
a sheaf F on C, F(X)|C denotes the tensor product, F ⊗OPn(X)|C.
Consider the last map in the first column of Equation 9. There is an associated
map of cohomology groups H1(C,−),
n−3
H1(C,Ωn−2| )→H1(C,ω ⊗( N )∨).
X C C ^ C/X
By Serre duality this is equivalent to,
n−3
H1(C,Ωn−2| )→Hom(H0(C, N ),C).
X C ^ C/X
10