Spaces of rational curves in complete intersections

0.25 MB·English

by  Roya Beheshti

SPACES OF RATIONAL CURVES IN COMPLETE
INTERSECTIONS
1
1 ROYABEHESHTI ANDN. MOHANKUMAR
0
2
Abstract. Weprovethatthespaceofsmoothrationalcurvesofdegree
n e inageneralcomplete intersection ofmultidegree(d1,...,dm)inPn is
Ja irreducible of the expected dimension if Pmi=1di < 23n and n is large
enough. This generalizes the results of Harris, Roth and Starr [9], and
9
isachievedbyprovingthatthespaceofconicspassingthroughanypoint
1
of a general complete intersection has constant dimension if Pmi=1di is
] small compared to n.
G
A
.
h
at 1. Introduction
m
Throughout this paper, we work over the field of complex numbers. For
[
a smooth projective variety X ⊂ Pn and an integer e ≥ 1, we denote

1 by Hilb (X) the Hilbert scheme parametrizing subschemes of X with
et+1
v
Hilbert polynomial et + 1, and we denote by R (X) ⊂ Hilb (X) the
7 e et+1
9 open subscheme parametrizing smooth rational curves of degree e in X. If
7 X = Pn, then R (X) is a smooth, irreducible, rational variety of dimension
e
3 (e+ 1)(n + 1) − 4. But already in the case of hypersurfaces in Pn, there
.
1 are many basic questions concerning the geometry of R (X) which are still
e
0
open. In this paper we address and discuss some of these questions, focus-
1
1 ing in particular on the dimension and irreducibility of Re(X), when X is a
: general complete intersection in Pn.
v
i To study the space of smooth rational curves in X, we consider the Kont-
X
sevich moduli space of stable maps M (X,e) which compactifies R (X)
0,0 e
r
a by allowing smooth rational curves to degenerate to morphisms from nodal
curves. These have certain advantages over the Hilbert Schemes for the
problems studied here. We refer the reader to [1, 5, 8, 9] for detailed discus-
sions of these moduli spaces and the comparison between them.
For every smooth hypersurface X ⊂ Pn of degree d, the dimension of
every irreducible component of M (X,e) is at least e(n+1−d)+n−4,
0,0
and if d ≤ n − 1, then there is at least one irreducible component whose
dimension is equal to e(n + 1−d) +n −4 (see Sections 2.1 and 6). The
number e(n + 1 − d) + n − 4 is referred to as the expected dimension of
M (X,e). If X is an arbitrary smooth hypersurface, M (X) (or even
0,0 0,0
R (X)) can be reducible and its dimension can be larger than expected
e
(see [4], Section 1). By a result of Harris, Roth, and Starr [9], if d < n+1
2
and X is a general hypersurface of degree d in Pn, then for every e ≥ 1,
1
2 ROYABEHESHTIANDN.MOHANKUMAR
M (X,e) isintegral oftheexpecteddimensionandhasonlylocalcomplete
0,0
intersection singularities. In this paper, we generalize this result to higher
degree hypersurfaces.
LetM (X,e)denotethemodulispaceof1-pointedstablemapsofdegree
0,1
e to X. In order to obtain the above mentioned result, Harris, Roth, and
Starr show that if d < n+1 and X is a general hypersurface of degree d in
2
Pn, then the evaluation morphism
ev : M (X,e) → X
0,1
is flat of relative dimension e(n+1−d)−2 for every e ≥1 ([9, Theorem 2.1
and Corollary 5.6]). It is conjectured that the same holds for any d ≤ n−1:
Conjecture 1.1 (Coskun-Harris-Starr[4]). Let X be a general hypersurface
of degree d ≤ n−1 in Pn. Then the evaluation morphism
ev : M (X,e) → X
0,1
is flat of relative dimension e(n+1−d)−2 for every e ≥ 1.
The above conjecture would imply the following:
Conjecture 1.2 (Coskun-Harris-Starr [4]). If X is a general hypersurface
of degree d ≤ n−1 in Pn, then for every e ≥ 1, M (X,e) has the expected
0,0
dimension e(n+1−d)+n−4.
Coskun and Starr [4] show that Conjecture 1.2 holds for d< n+4. When
2
d = n−1 and e ≥ 2, M (X,e) is reducible for the following reason. By
0,0
Lemma 6.2, M (X,e) has at least one irreduciblecomponent of dimension
0,0
2e+n−4 whose general point parametrizes an embedded smooth rational
curve of degree e in X. On the other hand, the space of lines in X has
dimension at least n−2, and therefore, the space of degree e covers of lines
in X has dimension ≥ (n − 2) + (2e − 2), thus M (X,e) has at least 2
0,0
irreducible components. It is expected that if X is general, then M (X,e)
0,0
is irreducible when d ≤ n−2, and R (X) is irreducible when d ≤ n−1 (see
e
[4, Conjecture 1.3]).
In this paper, we show:
Theorem 1.3. Suppose that X is a general hypersurface of degree d in Pn.
If
n−d
n−1 < ,
(cid:18) 2 (cid:19)
then the evaluation morphism ev : M (X,2) → X is flat of relative dimen-
0,1
sion 2n−2d.
A smooth rational curve in X is called free if its normal bundle in X is
globally generated. The proof of the above theorem is based on an analysis
of the space of non-free conics in X passing through an arbitrary point of
X. It seems quite plausible that the same approach can be applied to the
case of cubics or other higher degree rational curves to prove the flatness
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS 3
of ev when d and n satisfy the inequality of the theorem, but we have not
carried out all the details, and we restrict the discussion here to the case of
conics.
Theorem 1.3 along with the results of [9] gives the following:
Theorem1.4. IfX ⊂ Pn isageneral hypersurface of degree d< 2n,n ≥ 20,
3
then for every e ≥ 1,
(a) The evaluation morphism M (X,e) → X is flat and of relative
0,1
dimension e(n+1−d)−2.
(b) M (X,e) is an irreducible local complete intersection stack of ex-
0,0
pected dimension e(n+1−d)+(n−4).
The above results can be generalized to the case of general complete
intersections. Suppose that X ⊂ Pn is a general complete intersection of
multidegree (d ,...,d ) and set d = d +···+d . If n−m < n−d , then
1 m 1 m 2
the evaluation morphism ev : M (X,2) → X is flat of relative(cid:0)dim(cid:1)ension
0,1
2(n−d), and if d and n are in the range of Theorem 1.4, then M (X,e)
0,0
is an irreducible, local complete intersection stack of expected dimension
e(n+1−d)+n−3−m (Theorem 7.1).
Along the way to proving Theorem 1.3, we obtain the following result on
the space of non-free lines in general hypersurfaces.
Theorem 1.5. Let X be a general hypersurface of degree d in Pn, and let
p be an arbitrary point of X. For 1 ≤ k ≤ d−1, set
a+k+2
a = min a ≥ 0 | ≥ n .
k
(cid:26) (cid:18) k+1 (cid:19) (cid:27)
Then the family of lines l in X passing through p with h1(l,N (−1)) ≥ k
l/X
has dimension ≤ a .
k
In fact, we can modify the proof of the above theorem to say more in
special cases. For example, it follows from the proof of the theorem that if
n ≤ 5, then the space of non-free lines through any point of X is at most
zero dimensional. The proof shows that if there is a 1-parameter family of
non-freelines inX throughpparametrized by C ⊂ Pn−1, then n≥ 2+2dim
Linear Span(C) (see Proposition 6.5). Of course, a general hypersurface of
degree ≥ 3 in Pn, n ≤ 5, does not contain any 2-plane, so the dimension of
the linear span of C is at least 2. Note that for a general hypersurface X
of degree 3 ≤ d ≤ n−1, the non-free lines in X sweep out a divisor in X
(Proposition 6.3).
Acknowledgements. The first named author is grateful to Izzet Coskun
for many helpful conversations and for pointing out to us how to make our
original proof of Theorem 1.3 shorter. She also thanks Matt Deland and
Jason Starr for useful discussions.
4 ROYABEHESHTIANDN.MOHANKUMAR
2. Background and Summary
2.1. Preliminaries. Fix positive integers d ≤ ··· ≤ d , and set d :=
1 m
d + ··· + d . Let X be a smooth complete intersection of multidegree
1 m
(d ,...,d ) in Pn. The Kontsevich moduli space M (X,e) parametrizes
1 m 0,r
isomorphism classes of tuples (C,q ,...,q ,f) where
1 r
(1) C is a proper, connected, at worst nodal curve of arithmetic genus
0.
(2) q ,...,q are distinct smooth points of C.
1 r
(3) f : C → X is a map of degree e in the sense that the pull back
of the hyperplane section line bundle of X has total degree e on
C, which further satisfies the following stability condition: any ir-
reducible component of C which is mapped to a point by f has at
least 3 points which are either marked or nodes.
The tuples (C,q ,...,q ,f) and (C′,q′,...,q′,f′) are isomorphic if there is
1 r 1 r
anisomorphismg : C → C′ takingq toq′,withf′◦g = f. Themodulispace
i i
M (X,e)isaproperDeligne-Mumfordstack,andthecorrespondingcoarse
0,r
moduli space M (X,e) is a projective scheme. There is an evaluation
0,r
morphism
ev :M (X,e) → Xr
0,r
sending a datum (C,q ,...,q ,f) to (f(q ),...,f(q )). We refer to [1] and
1 r 1 r
[8] for constructions and basic properties of these moduli spaces.
The space of first order deformations of the map f with (C,q ,...,q )
1 r
fixed can be identified with H0(C,f∗T ). If H1(C,f∗T ) = 0 at a point
X X
(C,q ,...,q ,f), then f is unobstructed and the moduli stack is smooth at
1 r
that point. In particular, M (Pn,e) is smooth of dimension (n+1)(e +
0,r
1)+r−4 (see Appendix A of [14] for a brief discussion of the deformation
theory of M (X,e)).
0,r
Denote by C → M (Pn,e) the universal curve and by h : C → Pn the
0,0
universal map
C h // Pn
π
(cid:15)(cid:15)
M (Pn,e).
0,0
For any d ≥ 1, the line bundle h∗OPn(d) is the pullback of a globally gen-
erated line bundle and so it is globally generated. The first cohomology
group of a globally generated line bundle over a nodal curve of genus zero
vanishes, so by the theorem of cohomology and base change [10, Theorem
12.11], E := π∗h∗(⊕mi=1OPn(di)) is a locally free sheaf of rank de+m.
If s ∈ ⊕mi=1H0(Pn,OPn(di)) is a section whose zero locus is X, then π∗h∗s
is a section of E whose zero locus as a closed substack of M (Pn,e) is
0,0
M (X,e) ([9, Lemma 4.5]). The number
0,0
dimM (Pn,e)−(de+m)= e(n+1−d)+n−m−3
0,0
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS 5
is called theexpected dimension of M (X,e). Itfollows that thedimension
0,0
of every component of M (X,e) is at least the expected dimension, and if
0,0
the equality holds, then M (X,e) is a local complete intersection substack
0,0
of M (Pn,e).
0,0
Similarly, M (Pn,e) is a smooth stack of dimension (e+1)(n+1)−3,
0,1
and M (X,e) is the zero locus of a section of a locally free sheaf of rank
0,1
de+monM (Pn,e). ThereforeifdimM (X,e) = e(n+1−d)+n−m−2,
0,1 0,1
then it is a local complete intersection stack.
The number e(n+1−d)+n−3−m can be also obtained as an Euler
characteristic: if C is a smooth rational curve of degree e in X, and if N
C/X
denotes the normal bundle of C in X, then
χ(N ) = χ(T | )−χ(T ) = e(n+1−d)+n−m−3.
C/X X C C
2.2. Outline of proof of Theorem 1.3. Let X be a general hypersurface
of degree d in Pn, and consider the evaluation morphism
ev : M (X,e) → X.
0,1
Toshowthatev isflat,itsufficestoshowthateveryfiberofevhasdimension
e(n + 1 − d) − 2 (see the discussion in the beginning of Section 5). Let
e = 2, and assume to the contrary that there is a point p in X and an
irreduciblecomponentMofev−1(p)whosedimensionislargerthan2(n−d).
Itfollowsfrom[9,Theorem2.1]thatageneralmapparametrizedbyMisan
isomorphism onto a smooth conic passing through p. Let (P1,f : P1 → X)
be a general map parametrized by M, and let C denote the image of f. We
have
χ(N (−p)) = χ(T | (−p))−χ(T (−p)) = 2(n−d).
C/X X C C
By [11, II, Theorem 1.7] the Zariski tangent space to ev−1(p) at (P1,f)
is isomorphic to H0(C,N (−p)), therefore if dimM > 2(n − d), then
C/X
h0(C,N (−p)) > 2(n−d) and H1(C,N (−p))6= 0.
C/X C/X
There is a short exact sequence
0 → NC/X(−p)→ NC/Pn(−p)→ OC(d)(−p) → 0,
and we show that under the above assumptions, there is a subspace W of
codimension at most n−1 in H0(Pn,OPn(d)(−p)) such that for any w in W
and for a general stable map (P1,f : P1 →∼ C) parametrized by M, w| is
C
in the image of the map
ρ: H0(C,NC/Pn(−p))→ H0(C,OC(d)(−p))
obtained from the above sequence (Proposition 3.4). We then show in Sec-
tions 4 and 5 that if n< min n−d ,(d−1)⌊n−d⌋+1 , then the existence
2 2
(cid:16) (cid:17)
of such W implies that for a ge(cid:0)nera(cid:1)l (P1,f : P1 →∼ C) parametrized by M,
the map ρ is surjective. Applying the long exact sequence of cohomology
to the above short exact sequence we get H1(C,N (−p)) = 0, which is a
C/X
contradiction.
6 ROYABEHESHTIANDN.MOHANKUMAR
3. Deformations of rational curves
We fixafew notations fornormalsheaves first. If Y is aclosed subscheme
of a smooth variety Z, as usual we write N for the normal sheaf of Y in
Y/Z
Z. More generally, suppose that f : Y → Z is a morphism between quasi-
projective varieties and Z is smooth. Denote by T and T the tangent
Y Z
sheaves of Y and Z, and denote by N the cokernel of the induced map
f
T → f∗T . We refer to N as the normal sheaf of f. We may sometimes
Y Z f
write N instead to emphasize the range. If Y and Z are both smooth
f,Z
and f is generically finite then the exact sequence T → f∗T → N → 0
Y Z f
is exact on the left. If g : Z → T is another morphism to a smooth variety
T, then we get an exact sequence on Y of normal sheaves,
(1) N → N → f∗N → 0.
f gf g
Let B and X be smooth quasi-projective varieties. Supposethat π :Y →
B is a smooth projective morphism and denote by Y the fiber over b ∈ B.
b
Let F : Y → B ×X be a morphism over B such that the restriction of F
to every fiber of π is generically finite. Let f = F : Y → X and p (resp.
|Yb b B
p ) be the projections from B×X to B (resp. X). Notice that p ◦F = π
X B
andT isnaturallyisomorphictop∗ T ⊕p∗ T . Thus,wehaveanatural
B×X B B X X
map
(2) α: π∗T → N .
B F
If b ∈ B, then we have N | = N using the following commutative
F Yb f
diagram.
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 // TYb // f∗TX // Nf // 0
=
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 // TY|Yb // F∗TB×X|Yb // NF|Yb // 0
(cid:15)(cid:15) (cid:15)(cid:15)
NYb/Y = // π∗TB|Yb
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
Thus we get, restricting α, a map π∗T → N and since one has a
B,b f
natural map T → H0(Y ,π∗T ), we get a map,
B,b b B,b
(3) α :T → H0(Y ,N ).
b B,b b f
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS 7
3.1. Morphisms from P1 to general complete intersections. Letd ≤
1
··· ≤ d bepositiveintegers. Fortherestofthissection, wefixthefollowing
m
notation:
(1) H denotes the variety parametrizing complete intersections in Pn
which are of multidegree (d ,...,d ).
1 m
(2) U ⊂H×Pn denotes the universal family over H.
(3) π :U → H and π :U → Pn denote the two projection maps.
1 2
Remark 3.1. If h ∈ H and U ⊂ Pn is the corresponding scheme, then we
h
havethemapαh :TH,h → H0(Uh,NUh/Pn)asdefinedintheprevioussection.
(Easy to check that this is defined even though U → H is not smooth). This
map is an isomorphism for any h∈ H.
Suppose that B is an smooth irreducible quasi-projective variety and ψ :
B → H a dominant morphism. Let U ⊂ B ×Pn be the fiber product. Let
B
π : Y → B be a dominant morphism whose fibers are smooth connected
projective curves, and let F : Y → U be a morphism over B which is
B
generically finite on each fiber of π. We have an exact sequence using the
sequence (1),
(4) 0 → NF,UB → NF,B×Pn → F∗NUB/B×Pn → 0.
The natural map α :π∗TB → NF,B×Pn by composing gives a map
(5) β :π∗TB → F∗NUB/B×Pn.
The dominance of ψ and the above remark show that for a general point
b ∈ B and X the corresponding complete intersection subscheme of Pn, one
has a surjection TB,b → H0(X,NX/Pn).
Fix a point b ∈ B and let C ⊂ Y be the fiber over b. Let f = F and let
|C
X be the complete intersection scheme corresponding to b. Then the exact
sequence (4) specializes to
(6) 0 → Nf,X → Nf,Pn → f∗NX/Pn = ⊕mi=1f∗OX(di)→ 0
Proposition 3.2. If b is a general point of B, then the image of the pull-
back map
H0(Pn,⊕mi=1OPn(di)) → H0(C,⊕mi=1f∗OX(di))
is contained in the image of the map
H0(C,Nf,Pn) → H0(C,⊕mi=1f∗OX(di))
obtained from the above short exact sequence.
8 ROYABEHESHTIANDN.MOHANKUMAR
Proof. Since b is general, it is a smooth point of B, so we have α : T →
b B,b
H0(C,Nf,Pn) as before. We have a commutative diagram
TB,b αb // H0(C,Nf,Pn)
dψ
(cid:15)(cid:15)
(cid:15)(cid:15) (cid:15)(cid:15)
TH,[X] = H0(X,⊕mi=1OX(di)) // H0(C,⊕mi=1f∗OX(di))
and dψ is surjective since ψ is dominant and b is a general point of B, so
the result follows. (cid:3)
A curve C ⊂ Pn is called d-normal if the restriction map
H0(Pn,OPn(d′)) → H0(C,OC(d′))
is surjective for every d′ ≥ d.
Corollary 3.3. Suppose that X ⊂ Pn is general complete intersection of
multidegree (d ,...,d ), d ≤ ··· ≤ d . If C is any d -normal smooth
1 m 1 m 1
rational curve of degree e in X, then H1(C,N ) = 0. In particular,
C/X
R (X) is smooth at [C].
e
Proof. If f : P1 → C ⊂ X is an isomorphism, then H1(P1,Nf,Pn) = 0. So
applying the long exact sequence of cohomology to the sequence of normal
sheaves
0 → Nf,X → Nf,Pn → ⊕mi=1f∗OX(di) → 0,
the statement follows from Proposition 3.2. (cid:3)
Now, we make a further assumption that there exists a morphism φ :
B → U so that the composition with the map π : U → H is ψ. This is
1
equivalent to saying that we are given a section σ for the map U → B.
B
Next, for a point ([X],p) in the image of φ: B → U, we define a subspace
WX,p of H0(X,NX/Pn) as follows. For p ∈ Pn, let Bp = (π2 ◦φ)−1(p) with
the reduced induced structure, and let H ⊂ H be the closure of ψ(B ). If
p p
([X],p) is in the image of φ , then [X] ∈ H . We can identify T with
p H,[X]
H0(X,NX/Pn) and we defineWX,p ⊂ H0(X,NX/Pn)to beTHp,[X] underthis
identification.
AssumenowthatbisageneralpointofB andφ(b) = ([X],p). Sinceevery
complete intersection which is parametrized by H contains p, we have
p
WX,p ⊂ H0(X,NX/Pn(−p)).
Since ψ is dominant by our assumption, the codimension of H in H is at
p
most n. Thus WX,p is of codimension ≤ n in H0(X,NX/Pn), and it is of
codimension ≤ n−m in H0(X,NX/Pn(−p)).
We make a further assumption on F : Y → U that σ(B) ⊂ F(Y). Let
B
C be the fiber of π over b and f : C → X the restriction of F to C, so the
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS 9
image of f is a curve which passes through p. Let D = (f−1(p)) ⊂ C.
red
Then the image of the pullback map
WX,p → H0(C,f∗NX/Pn)
is contained in H0(C,f∗NX/Pn(−D)). Consider the short exact sequence of
normal sheaves (6) twisted with O (−D):
C
0→ Nf,X(−D)→ Nf,Pn(−D)→ f∗NX/Pn(−D)→ 0.
We claim that the image of the pull-back map
WX,p → H0(C,f∗NX/Pn(−D))
is contained in the image of the map
H0(C,Nf,Pn(−D)) → H0(C,f∗NX/Pn(−D))
obtained from the above short exact sequence. Since b is a general point,
and since the morphism B → H is dominant, the induced map on the
p p
Zariski tangent spaces T → T = W is surjective. Set
Bp,b Hp,[X] X,p
Y = π−1(B ),
p p
and let αb : TBp,b → H0(C,Nf,Pn) be the natural map corresponding to the
morphism Y → B ×Pn. Set
p p
Z = (F−1(B ×{p})) ⊂ Y .
p p red p
Then the fiber of Z over b is D. Let g : D → X be the restriction of f to
p
D, and let βb : TBp,b → H0(D,Ng,Pn)= H0(D,f∗TX|D) be the natural map
corresponding to the morphism Z → B ×Pn. Then β is clearly the zero
p p b
map. The claim now follows from the following commutative diagram
TBp,b NNNNNNαNNNNN // // WX,p ⊂ H0(X,NX/Pn)
&& (cid:15)(cid:15)
0 H0(C,Nf,Pn) // H0(C,f∗NX/Pn)
(cid:15)(cid:15) (cid:15)(cid:15)
f∗TPn|D // Nf,Pn|D.
We summarize the above discussion in the following proposition.
Proposition 3.4. For a general point ([X],p) in the image of φ, there is a
subspace
WX,p ⊂ H0(X,NX/Pn(−p))
of codimension at most n−m and an open subset B ⊂ φ−1([X],p) with the
0
following property: for every b ∈ B , if we denote the fiber of π over b by C,
0
the restriction of F to C by f : C → X, and the inverse image of f−1(p)
with the reduced induced structure by D, then for every w ∈ W , f∗w can
X,p
be lifted to a section of H0(C,Nf,Pn(−D)).
10 ROYABEHESHTIANDN.MOHANKUMAR
4. Conics in projective space
Let Hilb (Pn) denote the Hilbert scheme of subschemes of Pn with
2t+1
Hilbert polynomial 2t+1. In this section we prove the following:
Proposition 4.1. Suppose that p is a point of Pn and R ⊂ Hilb (Pn)
2t+1
is an irreducible projective subscheme of dimension r such that every curve
parametrized byRisasmoothconicthroughp. LetW ⊂ H0(Pn,OPn(d)(−p))
be a subspace of codimension c. If
r+1 r+1
c< min ,(d−1) +1 ,
(cid:18)(cid:18) 2 (cid:19) (cid:22) 2 (cid:23) (cid:19)
then for a general [C] ∈ R, and for every 2 ≤ k ≤ 2d, the image of the
restriction map
W → H0(C,O (d))
C
contains a section of O (d) which has a zero of order k at p.
C
We start with a lemma.
Lemma 4.2. If l is a line in Pn through p, then there is no complete one-
dimensional family of smooth conics through p all tangent to l.
Proof. Assume to the contrary that there is such a family B. By passing to
a desingularization we can assume that B is smooth. Let Y ⊂ B ×Pn be
the universal family over B and g :Y → Pn the projection map.
The point p gives a section σ of the family Y → B. Fix a point q 6= p
p
on l. Then for every conic C in the family, there is a unique line l′ 6= l
such that l′ passes through q and is tangent to C. Let σ be the section of
q
Y → B such that σ ([C]) is the point of intersection of C and l′. Since if
q
q and q are two distinct points of l, σ and σ are disjoint, Y ≃ B ×P1.
1 2 q1 q2
Since the section σ is contracted by g, by the rigidity lemma [13], g factors
p
through the projection map B ×P1 → P1. Thus the image of g should be
one-dimensional which is a contradiction.
Remark 4.3. The following argument shows that the set-theoretic map
σ defined in the proof of the above lemma is a section. If q 6= p on l,
q
B ×{q}∩Y = ∅. Thus the projection from q defines a morphism g : Y →
B×Pn−1. For any point b ∈ B, this map is just a map from a conic to a line
and thus g is a two-to-one map to its image. Let R ⊂ Y be the ramification
locus. Then, the map R → B is a double cover. But, B ×{p} ⊂ R and the
residual part is a section of Y → B and just σ .
q
(cid:3)
Corollary 4.4. Suppose that R ⊂ Hilb (Pn) is a closed subscheme such
2t+1
that every curve parametrized by R is a smooth conic through p. Then,
(1) dimR ≤ n−1.
(2) If the 2-planes spanned by the curves parametrized by R all pass
through a point q 6= p, then dimR ≤ 1.