Hilbert schemes of rational curves on Fano hypersurfaces

0.17 MB·English

by  Bin Wang

HILBERT SCHEMES OF RATIONAL CURVES ON FANO
HYPERSURFACES
B. WANG
JAN 3, 2015
Abstract. In this paper we try to further explore the linear model of the moduli of rational
5 maps. Our attempt yields following results. Let X ⊂ Pn be a generic hypersurface of degree h.
1 LetRd(X,h)denotetheopensetoftheHilbertschemeparameterizingirreduciblerationalcurvesof
0 degreedonX. Weobtainthat
2 (1)If4≤h≤n−1,Rd(X,h)isanintegral,localcompleteintersectionofdimension

n (0.1) (n+1−h)d+n−4.
a
J (2)Iffurthermore(h2−n)d+h≤0andh≥4,inadditiontopart(1),Rd(X,h)isalsorationally
connected.
4
2

] 1 Introduction
G
A WeworkoverthefieldCthroughout. HypersurfacesX ofprojectivespacePncan
. be classified into three different categories: (1) Fano, (2) Calabi-Yau, (3) of general
h
t type. In our previous papers [11], [12], [13], we conclude that, in all three categories,
a
the normal sheaves of rational curves on general hypersurfaces have vanishing higher
m
cohomology groups. This property is local. In this paper, we concentrate on the
[
global properties in first category, Fano hypersurfaces. In Fano case, we expect that

1 the“parameter”spacesofrationalcurvesonX hasapositivedimension,andsothere
v are plenty of rational curves that all have no obstruction. First let’s state the main
0
theorem.
7
0
6 Let Pn be projective space of dimension n over complex numbers C. Let
0
. (1.1) R (X,h) c:c X
1 d ⊂{ ⊂ }
0
denote the open set of the Hilbert scheme parameterizing irreducible rational curves
5
1 of degree d on X. This is a subscheme of the Hilbert scheme of rational curves of
: degree d in Pn.
v
i
X
Theorem 1.1. (Main theorem).
r (a) If 4 h n 1, R (X,h) is an integral, local complete intersection of
a ≤ ≤ − d
dimension
(1.2) (n+1 h)d+n 4.
− −
(b) Furthermore, if (h2 n)d+h 0 and h 4, R (X,h) is a rationally connected,
d
− ≤ ≥
integral, local complete intersection of dimension
(1.3) (n+1 h)d+n 4.
− −
Remark 1.1. A generic member of R (X,h) in the range of theorem 1.1 is
d
a smooth rational curve. This assertion is not included in the theorem. As far as
1
we know, there are elementary proofs of the existence for lines, but there are no
elementary proof for a general degree d. Clemens’ proof on quintic threefolds ([2]) is
one of them we know for general degree d. The importance of this remark is that in
the range 4 h n 1, J, Harris, M. Roth and J. Starr’s R (X) is a non-empty
d
≤ ≤ −
open set of our R (X,h) because R (X,h) will be proved to be irreducible.
d d
1.1 Related work
(1)Theresultsinourpreviouspapers[11],[12],[13]implythatR (X,h)forn 4
d
≥
is a reduced, local complete intersection of dimension
(n+1 h)d+n 4,
− −
where the negative (n+1 h)d+n 4 is interpreted as the Hilbert scheme being
− −
empty, and X could be a generic complete intersection. The main theorem in this
paper only addressesthe remainingparts,the irreducibility andrationalconnectivity
of R (X,h). It is clear that the irreducibility could only occur when h is relatively
d
small.
(2) Main theorem is an extension of results in two papers [6], [7] by J. Harris, J.
Starr et al, in which they initiated the study of the open set R (X) of the Hilbert
d
scheme parameterizing smooth rational curves of degree d. This is achieved through
a detailed analysis of Kontsevich’s moduli spaces of stable maps. They are followed
by the works of Beheshti, Kumar and others. See section 4 for the details.
(3) Our method is different from that in [6], [7]. This difference stems from the
beginning choice of “parameter” spaces of rational maps, i.e. a parameter space that
parametrizes families of rational maps. They used the Kontsevich’s moduli spaces
of stable maps, we use a linear model of it. Both methods analyze the structures
of “parameter spaces” that extend to the “boundaries”. The difference is rooted in
string theory’s approachof fields in“non-linear model” versus “linear model”.
1.2 Outline of the proof
In string theory, there are two different theories, “non linear sigma model” and
“gaugedlinear sigma model”. Kontsevich’s moduli space of stable maps is a starting
pointoftherigorous,mathematicaltheoryfor“nonlinearsigmamodel”. Ourresearch
focus onthe mathematicalstructures of fields in “gaugedlinear sigma model”, which
is also called a linear model of stable moduli in [4]. There is a filtration on this
modelwhichishelplesslysimple onitsown. Howeveritsinterplaywithhypersurfaces
is non trivial. The reason to use the linear model is that, the incidence scheme
of rational maps on generic hypersurface in the case of study, is a “mostly” smooth
subschemeofaprojectivespace. Oncetheschemeissmooth,everythingelsewillfollow
automatically. The linear model has advantages and disadvantages when comparing
withKontsevich’smodulispaceofstablemaps. Ourgeneralideain[11],[12],[13]and
this paper is to re-organizelocal coordinates of the linear model by breaking it down
tofiniteblocks,thentoanalyzeeachblockone-by-one. Thisisaccomplishedinsection
2. This method will turn the disadvantages of the linear model to its advantages.
Let S =P(H0( Pn(h))) be the space of all hypersurfaces of degree h.
O
Let
C(n+1)(d+1)
2
be the vector space,
(H0( P1(d))⊕n+1
O
whose open subset parametrizes the set of maps
P1 Pn
→
whose push-forward cycles have degree d. 1 Throughout the paper, we let
M =C(n+1)(d+1).
M has affine coordinates. The “gauged linear sigma model” uses the space M that
has a stratification of closed subvarieties,
(1.4) M =M M M = constant maps ,
d d 1 0
⊃ − ⊃···⊃ { }
where
(1.5) Mi = (gc0, ,gcn):g H0( P1(d i)),cj H0( P1(i)) .
{ ··· ∈ O − ∈ O }
ThisstratificationmakesitimpossibletoviewM asaspaceofmorphismsofthesame
degree d, i.e. M =Hom (P1,X).
d d
6
Let
Γ
be the incidence scheme
(1.6) (c,f) M S :c∗(f)=f(c(t))=0
{ ⊂ × }
Let Γ be the projection of the fibre of Γ over f to M.
f
The natural dominant rational map,
(1.7) Γ 9R9K R (X,h),
f d
reduces theorem 1.1 to showing that Γ is a rationally connected, integral variety of
f
the expected dimension. This rationalmap will be constructedand verifiedby the
R
results in [9], I 6.6.1, II 2.7. and in [10], prop. 0.9. We’ll discuss the details of this in
section 4.
Using this conversion,in the rest of the paper we concentrate on the scheme Γ .
f
Notice Γ has an induced filtration
f
(1.8) Γ (M Γ ) (M Γ ).
f d 1 f 0 f
⊃ − ∩ ⊃···⊃ ∩
Noticeby resultsin[9](mentionedabove) isregularonthe inverseofR (X,h)
d
R
because the rationalcurvesin R (X,h) are allirreducible. But it may not be regular
d
on the lower stratum of (1.4).
Then theorem 1.1 follows from the propositions on Γ below .
f
1TheautomorphismofP1 inducesaPGL(2)groupactiononP(C(n+1)(d+1)). Let
PGL(2)(c0)⊂P(Ch(d+1))
betheorbitofc0∈P(Ch(d+1)).
3
Proposition 1.2. If 4 h n, then for each d 1, the scheme
≤ ≤ ≥
(1.9) Γ M
f 0
\
is smooth.
Remark 1.2. The scheme Γ is singular at the points in M .
f 0
Proposition 1.3. If 4 h n 1, then for each d 1, the scheme
≤ ≤ − ≥
(1.10) Γ M
f 0
\
is connected.
Remark 1.3. When h=n our method failed to prove the connectivity of Γ .
f
Proposition 1.4. If h 4 and (h2 n)d+h 0, then the scheme
≥ − ≤
(1.11) Γ M
f 0
\
is a rationally connected, integral, complete intersection of M defined by
(1.12) f(c(t ))= =f(c(t ))=0,
1 hd+1
···
where t , ,t are any distinct points of P1.
1 hd+1
···
The propositions 1.3, 1.4 follow from the proposition 1.2 which follows from a
rather plausible, but difficult lemma
Lemma 1.5. Let be the Gauss map
G
(1.13) X (Pn) .
∗
→
Letc:P1 X beanon-constantregularmap(withan imageof anydegree). Assume
→
X is generic and h 4. Then for generic
≥
(t , ,t ) Symh(P1),
1 h
··· ∈
(c(t )), , (c(t ))
1 h
G ··· G
are linearly independent.
2 Smoothness of the linear model
Lemma 1.5 is the key to the results. Its proof lies in the heart of one difficult
question that is essential to many important problems in this area. In this paper we
wouldnotexplorethis difficultquestion,butreferittothe complete papers[11],[12],
and [13]. Let’s prove lemma 1.5.
4
Proof. of lemma 1.5: We proveit by a contradiction. Suppose there area generic
hypersurface X = div(f ) of degree h, a non-constant rational map c : P1 X ,
0 0 0 0
→
birational to its image, and h points
c (t ), ,c (t )
0 1 0 h
···
such that
(c (t )), , (c (t ))
0 1 0 h
G ··· G
are linearly dependent. Then
(2.1) dim( (c (t )) (c (t ))) n h+1
0 1 0 h
G ∩···∩G ≥ −
and for any vector α (c (t )) (c (t )),
0 1 0 h
∈G ∩···∩G
∂f
(2.2) 0 =0,
∂α|c0(t)
for all t P1. This is because
∈
(c (t )) (c (t )) (c (t))
0 1 0 h 0
G ∩···∩G ⊂G
for all t. Let α ,j =1, ,r =n h be a set of linearly independent vectors in
j
{ ··· − }
(c (t )) (c (t ))
0 1 0 h
G ∩···∩G
Then c lies on the hypersurfaces
0
∂f
(2.3) 0 =0,j =1, ,r.
∂α |c0(t) ···
j
(Notice f , ∂f0 are generic in the moduli of hypersurfaces). Hence it lies on the
0 ∂αj
intersection
∂f
(2.4) Y = 0 =0 X .
∩j{∂α }∩ 0
j
Next we are going to apply theorem 1.1 in [13]. We should elaborate the re-
quirements for the theorem. Let’s denote the sequence of hypersurfaces defining the
intersection Y by
∂f ∂f
(2.5) f ,f = 0, ,f = 0.
0 1 ∂α ··· r ∂α
1 r
There are three requirements for the proof of theorem 1.1 of [13]:
(1) the subvariety defined by f =0,0 j r is smooth of dimension n r 1
j
≤ ≤ − −
at c (P1);
0
(2)eachf ,j =0, ,risagenerichypersurface. Thisisdifferentfromtheactual
j
···
notion “generic complete intersection” which usually means that the point
(f , ,f )
(2.6) 1 ··· r ∈
H0( Pn(h)) H0( Pn(h 1)) H0( Pn(h 1))
O × O − ×···× O −
is generic.
5
(3) the first condition (1) says the subvariety Y at c (P1) is a local complete
0
intersection. The requirement is that dimension of the local complete intersection is
larger than or equal to 3.
Firsttwoconditionsaresatisfiedbecausef isgeneric. Byourassumptionh 4,
0
≥
we obtain that
(2.7) dim(Y)=h 1 3.
− ≥
The thid condition is also satisfied. Therefore by the theorem 1.1 in [13],
(2.8) H1(N )=0.
c0/Y
Next we apply H1(N )=0 to deduce an inequality. First let
c0/Y
(2.9) c∗0(TY)=OP1(a1)⊕···⊕OP1(adim(Y)).
Because H1(N )=0,
c0/Y
(2.10) a 1,j =1, ,dim(Y).
j
≥− ···
Becauseatleastone a is largerthanorequalto 2 (fromautomorphismsof P1 ), we
j
obtain that
(2.11) c (c (T )) dim(Y)+3= h+4.
1 ∗0 Y ≥− −
Now we use adjunction formula to find
(2.12) c1(c∗0(TY))=[n+1−(h+(n−h)(h−1))]d.
Now we apply the inequality h n 1 to obtain that
≤ −
(2.13) c1(c∗0(TY))≤−h+3
Then (2.11) becomes
(2.14) h+3 h+4.
− ≥−
This is absurd. Therefore c does not exist. The lemma 1.5 is proved.
0
Next we prove proposition 1.2:
Proof. The idea of the proof is similar to that in [11] or [12]. We are going to
choose affine coordinates for M and defining equations for Γ . Then use them to
f
calculate the Jacobian matrix of Γ . Let’s start with coordinates of M. We consider
f
a c Γ M . Let c be the normalization of c (P1). Then lemma 1.5 holds for c .
0 ∈ f\ 0 ′0 0 ′0
Let t ,t , ,t be those points in lemma 1.5., i.e.,
1 2 h
···
(c(t )), , (c(t ))
1 h
G ··· G
are linearly independent. Next we extend
(c(t )), , (c(t ))
1 h
G ··· G
6
to coordinates of Pn. That is to choose affine coordinates
z , ,z
0 n
···
of Cn+1 such that z = 0 for i = 1, ,h are exactly (c (t )). Next we choose
i 0 i
{ } ··· G
affine coordinates for M. In each copy H0( P1(d) of M, we express
O
d
cj(t)= ckjtk ∈H0(OP1(d))
kX=0
(for j-th copy) as
d
(2.15) c (t)= θk(t t )k.
j j − j
Xk=0
where t for j = 1, ,h are the those in lemma 1.5, and t = 0 if j is not in the
j j
···
interval [1,h]. The θk are affine coordinates for M. We would like to use coordinates
j
wk satisfying (a linear transformation of θk)
j j
wk =θk, k =0
(2.16) j j 6
(cid:26) w0 = d θk(t t )k.
j k=0 j ′− j
P
where t is a generic complex number.
′
Letthecorrespondingcoordinatesforthepointc beθ¯k. Nextwechoosedefining
0 j
equations of Γ at c . Consider the following homogeneous polynomials in Wk.
f 0 j
f(c(t))= n ǫ W0
′ j=0 j j
 ∂jf(Pc(t1)) j =1, ,d
(2.17)  ∂tj ···
···
∂jf(c(th)) j =1, ,d
 ∂tj ···
We claim that these polynomials define the scheme Γ .
f
Toseethis,weletcbeapointintheschemedefinedbythepolynomialsin(2.17).
Also let
hd
(2.18) f(c(t))= (c)ti.
i
K
Xi=0
Using an automorphism of P1, we may assume t =0. Then the equations
1
∂jf(c(t ))
1 =0,j =1, ,d
∂tj ···
imply that
(2.19) =0,1 i d.
i
K ≤ ≤
Then f(c(t)) satisfying the first set of equations
∂jf(c(t ))
1 =0,j =1, ,d
∂tj ···
7
becomes
d(h 1)
−
(2.20) f(c(t))= (c)+rd( (c)ti).
0 d+i
K K
Xi=1
Next we repeat the same process inductively for the term
d(h 1)
−
(c)ti
d+i
K
Xi=1
to obtain all = 0,i 1. At last = 0 because f(c(t)) = 0. Hence c Γ . To
i 0 ′ f
K ≥ K ∈
prove the proposition it suffices to show that the Jacobian matrix of
∂jf(c(t )) ∂jf(c(t ))
f(c(t′))=0, 1 =0, h =0
∂tj ∂tj
withrespecttothevariableswk hasfullrank. (Notewk arethecoordinatesforc). Let
j j
αk be the variables for T M with the basis ∂ . Consider the subspace V of T M
j c0 ∂wk T c0
j
defined by αk = 0 = αk where k = 0 and l is not one of 1, ,h. Then T Γ V
0 l 6 ··· c0 f ∩ T
consists of all α V satisfying
T
∈
∂f(c0(t′)) =0
∂α
 ∂j+1f(c0(t1)) =0 j =1, ,d
(2.21)  ∂tj∂α ···
···
∂j+1f(c0(th)) =0 j =1, ,d
 ∂tj∂α ···
We start this with h equations in (2.21) in the second derivatives. They are
equivalent to the equations
∂f(c (t )) ∂f(c (t ))
0 1 0 h
(2.22) = = =0
∂α ··· ∂α
1 1
where
α span(α1, ,α1).
1 ∈ 1 ··· h
By the lemma 1.5, we know that
∂f(c (t ))
0 i =δj
∂α1 i
j
where δj = 0 for i = j and δi = 0. Then the equations (2.22) implies α1 = 0,j =
i 6 i 6 j
1, ,h. Next step is to consider another h equations in third derivatives.
···
∂3f(c (t )) ∂3f(c (t ))
0 1 0 h
(2.23) = = =0
∂t2∂α ··· ∂t2∂α
Because α1 =0,j =0, ,n, we simplify (2.23) to
j ···
∂f(c (t )) ∂f(c (t ))
(2.24) 0 1 = = 0 h =0.
∂α2 ··· ∂α2
1 h
8
Then we use lemma 1.5 to obtain that
(2.25) α2 = =α2 =0.
1 ··· h
Recursively we obtain that the solution to the system of linear equations (2.21) is all
αk,j =1, ,h,k=0, ,d satisfying
j ··· ···
n ∂f(c0(t′)) =0
(2.26) i=0 ∂α0j
αk =0,Pj =1, ,h,k=1, ,d.
j ··· ···
Thismeansthatthesetofsolutionstotheequations(2.21)hasdimensionh 1. Thus
−
the rank of Jacobian matrix of Γ at c is hd+1, i.e. it has full rank. Hence Γ is
f 0 f
smooth at c whenever c is a non-constant.
0 0
This completes the proof.
2.1 Connectivity and a version of “bend and break”
This section will prove proposition 1.3. In last section we proved that
Γ M
f 0
\
is a smooth variety of dimension
(n+1)(d+1) (hd+1).
−
Toshowitisirreducible,itsufficestoshowitisconnected. Theideaoftheproofisto
connect a generic point of Γ to a point in the lower stratum. Then by the induction
f
it is connected to a point parametrizing the multiple of lines. This is our version of
“bend-and-break”.2 Let Γ be an irreducible component of Γ . Assume d 2.
′f f ≥
Then
(2.27) dim(Γ )=(n+1)(d+1) (hd+1)=(n+1 h)d+n
′f − −
Let
(2.28) Md−1 = P1(d 1)⊕n+1.
O −
We should note that M C Md 1. It has a similar stratification
d 1 −
− ≃ ×
(2.29) Md−1 =Mdd−11 ⊃Mdd−21 ⊃···⊃M0d−1 ={constant maps},
− −
where
Mid−1 ={(gc0,···,gcn):g ∈H0(OP1(d−1−i)),cj ∈H0(OP1(i))}.
Then every irreducible components of Γ M is isomorphic to an irreducible
′f ∩ d−1
component of
(2.30) C×Γdf−1,
2 Our“bend andbreak”failswhenh≥n. Thefailureisduetoapotential existence ofcertain
irreduciblecomponents. Butwedon’thaveanexampleofsuchfailure.
9
where Γdf−1 is defined to be
(2.31) c Md 1 :c f ,
−
{ ∈ ⊂ }
and C is an affine open set of P(H0( P1(1))). Notice
O
(2.32) dim(Γdf−1∩M0d−1)=d+n−1
dim(Γdf−1)=(n+1−h)(d−1)+n
Because h n 1, d 2,
≤ − ≥
(2.33) dim(Γdf−1)>dim(Γdf−1∩M0d−1).
Theinequality(2.33)holdsforeverycomponentsofΓdf−1. Thereforeeverycomponent
of Γ M contains a non-constantc. Thus inside smooth locus of Γ , every point
′f ∩ d−1 f
is connected to a point in the lower stratum. Then by the induction it suffices to
prove that the second lowest stratum Γ (M M ), which consists of all maps that
f 1 0
∩ \
correspond to lines, is connected. By the classical result of Fano variety of lines, this
is correct. More precisely
Γ M
f 1
∩
is isomorphic to
Cd 1 Γ1
− × f
where Γ1 is the same as (2.31) with d=2, and Cd 1 is an affine open set of
f −
P(H0( P1(d 1))).
O −
Then it suffices to prove
Γ1
f
is irreducible. The image of Γ1 under the rationalmap is just an open set of Fano
f R
variety F(X) of lines on the generic hypersurface X = f = 0 . It is connected by
{ }
the classical result (see theorem 4.3, [9]). Therefore the proposition 1.3 is proved.
3 Rationally connectedness
Proof. Note
(3.1) P(M )
0
is a smooth subvariety of P(n+1)d+n, with dimension
d+n 1.
−
Choose two generic planes V ,V in P(n+1)d+n with dimensions
top bott
nd 1,n+d
−
10