Lines on Fano 3-folds according to Shokurov
Miles Reid
0
Introduction and the main theorem
This paper is a reconstruction of the proof in [Sh] of the following result:
Theorem Let V be a complete 3-fold with ample anticanonical divisor −KV .
Then just one of the following holds:
(i) V contains a line;
(ii) index V ≥ 2;
(iii) V ∼
= P1 × P2 .
In this section, after some definitions, I state the main intermediate steps
Lemma 2.2, Lemmas 2.3–2.4 and Lemma 2.50 on which Shokurov’s proof is
based, and then give his proof using them.
I want to emphasise that not only the overall scheme of the proof, but
also many of the essential details of this paper have been lifted from [Sh] –
although in many cases I have been able to get rid of some of Shokurov’s
extraordinarily involved proofs, and deduce stronger corollaries. I am now
quite sure that [Sh] contains no serious errors or gaps, so that it contains a
complete – if somewhat elephantine – proof.
This paper had its origins in a review of [Sh] for Math Reviews. Conversations with Bob Friedman, Mark Levine, Ron Livné and David Mumford at
the August 1980 session of the Montreal/Nato S.M.S. impressed on me the
desirability of a laundry job on [Sh]. This paper was written during a month’s
stay at the Mittag-Leffler Institute in Sep 1980, whom I would like to thank
for a travel grant. I also thank Fabrizio Catanese for discussions.
Conventions
A Fano 3-fold is a nonsingular projective 3-fold V with ample −KV . The
index of V is that integer r such that 1r KV ∈ Pic V is primitive. The Fano
3-folds of index ≥ 2 are well understood [Isk, Part I, 4.2]; the varieties of
1
index 1 for which |−KV | has a base locus [Isk, Part I, 3.3], the hyperelliptic
ones for which ϕ−KV is not an isomorphism, [Isk, Part I, 7.2], and the trigonal
varieties for which the anticanonical image V2g−2 ⊂ Pg+1 is not an intersection
of quadrics [Isk, Part I, 2.5] are well known. All these either contain pencils
of del Pezzo surfaces, or are weighted complete intersections, and it is not
difficult to show that they have lines.
A line (respectively conic) C ⊂ V is an irreducible curve C for which
−KV · C = 1 (respectively 2). A Veronese surface (respectively scroll Fn;r )
in V is a surface S ∼
= P2 with OS (−KV ) ∼
= OP2 (2) and OS (−S) ∼
= OP2 (1)
∼
1
(O⊕O(−n))
=
F
,
with
O
(−K
)
O
(respectively S ∼
P
((n+r)A+B),
= P
n
S
V =
Fn
where A is the fibre of Fn → P1 , and B the negative section: B 2 = −n,
AB = 1, A2 = 0). We will be concerned especially with F0;2 and F2;1 , and
these will be referred to collectively as quartic scrolls. These surfaces are said
to be linearly normal if H 0 (OV (−KV )) H 0 (OS (−KV )); I will usually not
make this assumption.
I will say that a surface S ⊂ V is a surface of degree n − 1 spanning Pn
(written S = Sn−1 ⊂ Pn ) if deg S = (−KV )2 S = n−1, and H 0 (OV (−KV )) H 0 (OS (−KV )) with h0 (OS (−KV )) = n + 1. Under ϕ−KV , S goes into one of:
P2 , quadric ⊂ P3 , rational normal scroll (including cones),
normally embedded Veronese surface.
I always assume that |−KV | is free. If X ∈ |−KV | is a general element, and
C = X1 ∩X2 with Xi ∈ |−KV | general, then X is a K3 surface and C a curve,
with |−KV ||C the complete canonical system; C ⊂ X ⊂ V will be referred to
as general curve and K3 sections. By [S-D, p. 611], a complete linear system
|D| on X without fixed components has no base point; a simple argument
shows that the same holds for V provided that H 0 (OV (D)) H 0 (OX (D)).
I will use the shorthand (read “maps onto”) to mean that a linear map
is surjective. I will also write
|−KV | 3 D1 + D2 ,
with Di > 0
to mean that |−KV | contains a reducible divisor, and
|−KV | 63 D1 + D2 ,
with Di > 0
to mean that every D ∈ |−KV | is irreducible.
Plan of proof
Shokurov’s proof is based on the following results 2.2–2.5:
2
Lemma 2.2 ([Sh, §8], and Section 1 below) Assume that −KV is ample, and that |−KV | 3 D1 + D2 , with Di > 0. Then one of the following
holds: (i–iii) of the theorem, or
(iv) V contains a Veronese surface.
The following results are similar to those used in the Fano–Iskovskikh
method of projections and multiple projections from suitable loci in V .
Lemma 2.3–2.4 ([Sh, §§6–7], and Section 3 below) Assume that −KV
is very ample, and that V = V2g−2 ⊂ Pg+1 is an intersection of quadrics.
Suppose that |−KV | 63 D1 + D2 with Di > 0.
(A) Let P ∈ V be a point not lying on any line or conic. Construct the
blowup σ : W → V of P ∈ V , and set σ −1 P = S. Then −KW =
σ ∗ (−KV ) − 2S is again very ample.
(B) Let q ⊂ V be a conic not meeting any lines of V , and with NV |q ∼
=
OP1 ⊕ OP1 or OP1 (1) ⊕ OP1 (−1). Construct the blowup σ : W → V of
q ⊂ V , and set S = σ −1 q. Then −KW = σ ∗ (−KV ) − S is again very
ample.
Note that in (A), W ⊃ S = Veronese surface; in (B), W ⊃ S = quartic
scroll F0;2 or F2;1 .
0
Remark 1 The projected variety W = W2g0 −2 ⊂ Pg +1 is a Fano 3-fold with
g 0 = g − 4 or g − 3 in the two cases, and rank Pic W ≥ 2; such varieties do not
exist for g 0 ≤ 5. Hence proving Lemma 2.3–2.4 in the cases g = 6, 7 and 8 is
equivalent to proving that every conic meets a line and that every point is on
a line or conic.
Remark 2 See [Isk, Part II, 4.3] for information on conics; we need to
know that if q is a conic from a covering family then NV |q ∼
= OP1 ⊕ OP1
or OP1 (1) ⊕ OP1 (−1).
Lemma 2.50 ([Sh, §5], and Section 2 below) Let W be a 3-fold having
very ample −KW , and suppose that W ⊃ S = Veronese surface or quartic
scroll. Make the further assumption
|−KV | 63 D1 + D2 + nS,
with Di > 0 and S 6⊂ Di
(irred mod S.)
Then the linear system |E| = |−KW − 2S| is free, with dim ϕE (W ) = 3.
Note that Lemma 2.50 requires no hypothesis on lines of W .
The key result follows at once from Lemma 2.50 :
3
Proposition 2.5 Let W be a 3-fold with very ample −KW , and suppose that
W contains a Veronese surface or a quartic scroll. Then
|−KV | 3 D1 + D2 + nS,
with Di > 0 and S 6⊂ Di .
Proof The conclusion of Lemma 2.50 is impossible. For by the Grauert–
Riemenschneider vanishing theorem [G–R, p. 263], H i (OW (−E)) = 0 for
i = 1, 2. Hence H 1 (OE ) = 0 for E ∈ |E|, and χ(OE ) > 0. This is in
contradiction with the values
(
−2 if S = Veronese surface,
χ(OE ) =
−1 if S = quartic scroll,
calculated as follows: if H ∈ |−KW |, then χ(OH ) = 2 follows from
0 → OW (KW ) → OW → OH → 0.
Now if Ei ∈ |H − iS| for i = 1, 2, then OS (−Ei+1 ) = OS (KS + iS), and the
exact sequences
0 → OW (KW + (i + 1)S) → OW → OEi+1 → 0
and
0 → OW (KW + iS) → OW (KW + (i + 1)S) OS (KS + iS)
give χ(OE1 ) = 1 and χ(OE2 ) = −2 or −1 in the two cases. Q.E.D.
Remark Lemma 2.50 could perhaps also be used to bring the hypotheses of
Lemma 2.3–2.4 into contradiction with the results of Mori [M], which imply
that V has a large family of rational curves of degree ≤ 4.
Corollary If |−KV | 63 D1 + D2 with Di > 0 then every point of V lies on
a conic or a line, and every conic meets a line.
Proof of the Theorem Compare [Sh, §2]. We get a contradiction by
assuming that rank Pic V is minimal among V with
−KV is ample; and none of (i–iii) hold.
(∗)
First of all, −KV is very ample and V = V2g−2 ⊂ Pg+1 is an intersection of
quadrics, since otherwise V is known to contain a line. Now |−KV | 3 D1 +D2
with Di > 0, for otherwise, according as V is covered by conics or not, one of
the constructions (A) or (B) of Lemma 2.3–2.4 is applicable, to give a blown
up variety W satisfying the hypothesis of Proposition 2.5. Since σ ∗ (−KV ) =
4
−KW + εS (with ε = 2 or 1), the decomposition given by Proposition 2.5 also
gives |−KV | 3 D1 + D2 with Di > 0.
Now use Lemma 2.2; by (∗), V must contain a Veronese surface S. Since
OS (−S) ∼
= OP2 (1), V is the blowup σ : V → V1 of a nonsingular point P ∈ V1 ,
with σ −1 (P ) = S. Now because −KV = σ ∗ (−KV1 ) − 2S, we see that −KV1 is
ample on V1 . Obviously V1 cannot contain lines, V1 ∼
6= P1 × P2 and index V1 =
1, since otherwise the blowup V would contain curves C with −KV C ≤ 1,
or would itself have index ≥ 2. Thus V1 also satisfies (∗), but has smaller
rank Pic V1 . This contradiction proves the theorem.
1
Proof of Lemma 2.2
Assume that |−KV | is free and ample, and that |−KV | 3 D1 +D2 with Di > 0.
The proof of Lemma 2.2 consists of showing that every such V falls under one
of the following cases (a–e), and then reading off (i–iv) of Lemma 2.2 from
properties of these varieties.
(a) V has a free pencil |S| of del Pezzo surfaces.
(b) V contains a Sn−1 ⊂ Pn .
(c) V → P2 is a conic bundle.
(d) index V ≥ 2.
(e) V is the blowup of P3 or of the nonsingular quadric Q ⊂ P4 in a nonsingular curve C.
Remark The thorough study of each of these classes of varieties presents a
challenging problem.
(a) The study of 3-folds with a pencil of del Pezzo surfaces is begun in Step 4
below – for the purpose of proving Lemma 2.2, it is sufficient to deal with
the cases of surfaces of degree 8 and 9, since the others contain lines.
On the other hand, Fano 3-folds with pencils of del Pezzo surfaces of
degrees 1, 2 or 3 have been treated in [Isk], since for them either |−KV |
has a base locus [Isk, Part I, 3.3 and the erratum in the introduction
of the English translation of Part II], or V is one of the hyperelliptic or
trigonal varieties [Isk, Part I, 7.2 and Part II, 2.5]. In each case there
are one or two families other than the product P1 ×(del Pezzo surfaces).
(b) The surfaces Sn−1 ⊂ Pn are always exceptional: the scrolls arise from a
blowup σ : V → V1 , where V1 is another Fano 3-fold, and the centre of
the blowup is a curve C ⊂ V1 not meeting any lines of V1 ; similarly, the
5
Veronese surface arises by blowing up a point not on any line or conic.
P2 and Q ⊂ P3 can also be obtained in this way, where V1 is a “Fano
3-fold with Mori singularities”; for example, a V with planes can be
obtained by blowing up X4 ⊂ P(13 , 22 ), a variety of index 3/2 (compare
Dëmin [D, Theorem 1, 1B]), and Q by blowing up a double point of a
cubic. In each case V contains only lines in the exceptional surface.
Conjecture A Fano 3-fold is covered by conics, and contains an ample
divisor made up of scrolls of lines, provided that V is not one of the following
(or a blowup of one of the following):
• a variety of index ≥ 2;
• P1 × P2 ;
• a variety containing a plane.
There is a breath-taking analogy with del Pezzo surfaces.
(d) These are known [Isk, Part I, 4.2].
(e) Obviously accessible.
(c) This class is huge (see Sarkisov [S1] and [S2]).
Problem 1 Do there exist Fano 3-folds V such that rank Pic V ≥ 2, but
|−KV | 63 D1 + D2 with Di > 0, and V contains no contractible surfaces?
Problem 2 Classify the 3-folds V for which −KV = rH + D with r ≥ 1,
and D > 0; for surfaces we find again our old friends Sn−1 ⊂ Pn .
Step 1 Assume that V has no free pencil. Let H ∈ |−KV | and let Γ be a
component of H of minimal degree. Then |Γ| is free provided that Γ is not
an Sn−1 ⊂ Pn (for some n ≤ g).
Proof Write Γ + D = H ∈ |−KV |. I can obviously assume that D 6= 0.
Let C ⊂ X ⊂ V be general curve and K3 sections. Both Γ and Γ ∩ X are
irreducible, so H 1 (OV (−Γ)) = H 1 (OX (−Γ)) = 0, and
H 0 (OV (D)) H 0 (OX (D)) H 0 (OC (D)).
I claim that unless Γ = Sn−1 ⊂ Pn , |D| has no fixed components, and is hence
free. This follows from the following, which will be good fun for the reader:
6
Exercise ([Sh, 3.23]) Deduce the following from Clifford’s theorem: let
d > 0 be a special divisor on a curve C; then
deg(fixed part of |KC − d|) ≤ deg d,
with equality if and only if
either deg d = g − 1 and h0 (OC (d)) = 1,
or C is hyperelliptic, deg d = n − 1 and h0 (OC (d)) = 1 for some n ≤ g.
Thus |D| can only have a fixed component if Γ = Sn−1 ⊂ Pn , and is free
otherwise. By Bertini’s theorem, and the assertion that |D| is not composed
of a pencil, |D| is irreducible. Hence H 1 (OV (−D)) = 0, and
H 0 (OV (Γ)) H 0 (OX (Γ))
H 0 (OX (−KV )) H 0 (OX∩Γ (−KV )).
The assumption that Γ is fixed implies that Γ ∩ X is a −2-curve on X, hence
∼
= P1 , and is normally embedded by ϕ−KV . Thus Γ = Sn−1 ⊂ Pn . Q.E.D.
Step 2 I now suppose that |Γ| is irreducible, dim |Γ| ≥ 2, and H 0 (OV (Γ)) H 0 (OX (Γ)); if V has no free pencil and no Sn−1 ⊂ Pn , then every component
of every H ∈ |−KV | has this property.
The restriction |Γ||X is still irreducible, and of dimension ≥ 2. Hence
Γ2 X > 0; also |Γ| is free.
Proposition Suppose that every Γ ∈ |Γ| is irreducible, and dim ϕΓ (V ) = 2.
Then ϕΓ : V → P2 is a conic bundle. If furthermore ϕΓ has no reducible fibres
f3 (the blowup of a point in P3 ), or V contains lines.
then V ∼
= P1 × P2 or P
f3 has index 2.
Note that P
Proof ϕΓ (V ) certainly contains ϕΓ (X), which by [S-D, §5, p. 616] is either
birational to X, or is an Sn−1 ⊂ Pn . Now ϕΓ : V → X is obviously impossible,
so that if ϕΓ (V ) is 2-dimensional, it is Sn−1 ⊂ Pn . Now n = 2, for otherwise
|Γ| would contain reducible divisors: thus ϕΓ : V → P2 . If f is the fibre, then
−KV · f = 2, and f ∼
= P1 ; if every fibre is irreducible then V = P(ξ), with ξ
a rank 2 vector bundle over P2 . The last assertion comes from the following
lemma, using Van de Ven’s result on uniform bundles [VdV].
Lemma Let V = P(ξ) → P2 , with ξ a rank 2 vector bundle. Then if V is a
Fano 3-fold, for every line l ⊂ P2 , ξ |l ∼
= OP1 (a1 ) ⊕ OP1 (a2 ) with |a1 − a2 | ≤ 2;
if V contains no lines then |a1 − a2 | ≤ 1.
7
Proof (See also Dëmin [D]). Restricting over a line l ⊂ P2 , we get
F ⊂ V
↓
↓
l ⊂ P2
F ∼
= Fa with a = |a1 − a2 |.
We have KF = (KV + F )|F and OF (F ) = ϕ∗ Ol (1). If B ⊂ Fa is the negative
section, then B 2 = −a, KF · B = a − 2, and KV · B = a − 3. Q.E.D.
If |a1 − a2 | ≤ 1 for every line then Van de Ven’s result on uniform bundles
implies that ξ = O ⊕ O or O ⊕ O(1), giving the varieties indicated.
Step 3 Now suppose that none of the statements (a–d) at the beginning of
this section hold for V .
Then for every H ∈ |−KV |, and every divisor D with 0 < D < H, |D|
is free with D3 > 0. Let Γ be of minimal degree among such divisors, and
let r be the maximum integer such that H 0 (OV (−KV − rΓ) 6= 0. Then the
hypotheses of the next result are all satisfied.
Theorem Let V be a 3-fold, −KV = rΓ + F , with r ≥ 1, and suppose that
every Γ ∈ |Γ| is irreducible, |Γ| is free with Γ3 > 0 and H 0 (OV (Γ − F )) =
H 0 (OV (F − Γ)) = 0.
Then either r ≤ 2, Γ3 = 1, ϕΓ : V → P3 , or r = 1, Γ3 = 2, ϕΓ : V →
Q ⊂ P4 , with Q a nonsingular quadric; in either case ϕΓ is a small birational
morphism.
Furthermore, if V is a Fano 3-fold, then ϕD is a blowup of P3 or Q in a
nonsingular (possibly disconnected) curve.
Remark 3 A birational morphism ϕ : X → Y is small if dim ϕ−1 ≤ 1 for
every P ∈ Y . Recall that [Danilov] has proved that if X and Y are smooth and
projective, then every small birational morphism is a composite of blowups
in smooth codimension 2 centres.
Proof Let X ∈ |Γ + F | = |−KV − (r − 1)Γ| be a general element; X is a
nonsingular surface and −KX ∼ (r − 1)Γ|X .
Now we have exact sequences
0 = H 0 (OV (Γ − F )) → H 0 (OX (Γ − F )) → H 1 (OV (−2F )) = 0
0 = H 0 (OV (F − Γ)) → H 0 (OX (F − Γ)) → H 1 (OV (−2Γ)) = 0,
where the H 1 vanish because each of |2F | and |2Γ| contain irreducible divisors.
8
Now H 0 (OX (F − Γ)) = 0 certainly implies that H 0 (OX (F − rΓ)) = 0, so
that by Serre duality on X we get H 2 (X, OX (Γ − F )) = 0. Thus Riemann–
Roch on X gives
χ(OX ) + 12 (Γ − F )(rΓ − F )X = χ(OX (Γ − F )) ≤ 0.
(+)
Now let C = Γ1 ∩ Γ2 and D = F1 ∩ F2 be the general curve sections, with
Γi ∈ |Γ| and Fi ∈ |F | general elements; by hypothesis Γ3 > 0 and F 3 > 0, so
that C and D are irreducible curves, with genus p(C) and p(D) given by the
adjunction formula
2p(C) − 2 = Γ2 −F − (r − 2)Γ
2p(D) − 2 = F 2 (F − rΓ).
Substituting in the inequality (+) gives
2p(C) − 2 + 2(r − 1)D3 + 2p(D) − 2 ≤ −χ(OX ).
If r ≥ 2 then χ(OX ) = 1, and p(C), p(D) ≥ 0 force r = 2, Γ3 = 1.
If r = 1 then X ∈ |−KV | is a K3 surface, so that χ(OX ) = 2, and
p(C) = p(D) = 0; but H 1 (OV ) = H 1 (OΓ ) = 0, so that
H 0 (OV (Γ)) H 0 (OΓ (Γ)) H 0 (OC (Γ));
this implies that ϕΓ embeds each of the curves C as a rational normal curve.
Hence ϕΓ is birational, and ϕΓ (V ) is a 3-fold of degree n − 2 in Pn . Now
n = 3 or 4, for otherwise we contradict the hypothesis that every Γ ∈ |Γ| is
irreducible. Thus in either case, we have a birational morphism
ϕ Γ : V → P3
or ϕΓ : V → Q ⊂ P4
with Q a nonsingular quadric.
The morphism ϕΓ does not contract any surface S ⊂ V to a point, for
otherwise |Γ − S| =
6 ∅, contradicting the irreducibility of every Γ ∈ Γ. Hence
ϕΓ is small.
The last assertion is obtained by using Danilov’s theorem (it also follows
σ1
from [Sh, 4.12]): let V1 −→
V2 → · · · → Vn be any sequence of blowups with
nonsingular codimension 2 centres. If the centre of σ1 meets the exceptional
locus of σn ◦ · · · ◦ σ2 , it has to meet some line whose proper transform on V1
is a curve C with CKV1 ≥ 0. This proves the theorem. Q.E.D.
Note that ϕΓ cannot be an isomorphism – for then index V ≥ 2; and the
exceptional scroll consists of lines
9
Step 4 (preliminary version) Suppose that V = V2g−2 ⊂ Pg+1 is an intersection of quadrics, and that |S| is a free pencil on V , necessarily of del Pezzo
surfaces. Then
(i) every reducible fibre S ∈ |S| contains a component Γ = Sn−1 ⊂ Pn ;
(ii) if every element of |S| is irreducible, and if the general element has no
lines, then V is one of the following 4 varieties:
1. V = P1 × P1 × P1 ,
2. F1 × P1 (with F1 → P1 defining the pencil),
3. P1 × P2 or
4. F(1, 0, 0) (the blowup of P1 ⊂ P3 ).
The final version will contain a proof of the following result, strengthening
(i) while removing its hypotheses:
Theorem Let f : X → C be a morphism of a smooth 3-fold to a curve, and
suppose that −KX is relatively ample for f . Then every component Γ of a
reducible fibre is a relative Sn−1 ⊂ Pn ; that is, deg Γ = (−KX )2 Γ = n − 1,
and f∗ OX (−KX ) H 0 (OΓ (−KX )) with h0 (OΓ (−KX )) = n + 1.
The proof is based on
e for which −Ke is a “big” positive
(a) a study of nonsingular surfaces Γ
Γ
divisor, and
(b) numerical connectedness of the fibres.
Proof deg S ≤ 9, so that every reducible S contains a component of degree
≤ 4. (i) now follows from the fact that V is an intersection of quadrics.
Now suppose that every S ∈ |S| is irreducible, and that the general element has no lines, that is, S ∼
= P1 × P1 or P2 .
Replacing the given embedding by ϕ−KV +S , I can assume that every S ∈
|S| is linearly normal and is a Sn−1 ⊂ Pn embedded by −KS . The cone is
excluded by the nonsingularity of V . P1 × P1 and F1 cannot fit together in a
smooth family1 since they are not diffeomorphic (alternatively, argue on the
deformation of l ⊂ F1 ). Thus every S ∈ |S| is P1 × P1 or P2 .
1
Erratum, added in 1982: The treatment given here does not consider the possibility
that |S| is a pencil whose general member is P1 ×P1 , but which has singular fibres isomorphic
to the quadric of rank 3 Q ⊂ P3 . In this case X is a quadric bundle over P1 ; I have not as
yet checked that every Fano 3-fold having such a pencil satisfies one of (i–iv) of Lemma 2.2.
10
Lemma If f : X → C is a morphism to a curve, with every fibre ∼
= P1 × P1 ,
2
and if H (X, OX ) = 0 then X ∼
= P(ξ1 )×C P(ξ2 ), with ξi rank 2 vector bundles.
If every fibre is Pn then X ∼
= P(ξ), with rank ξ = n + 1.
Proof The Pn case is standard, and the usual proof can easily be modified
to give the P1 × P1 case.
Now one sees easily that the 4 listed varieties are the only Fano 3-folds of
this type. For example, if B1 ⊂ Fa1 and B2 ⊂ Fa2 are the 2 negative sections,
then
l ⊂ Fa1 ×P1 Fa1 , NFa1 ×P1 Fa1 |l ∼
= O(−a1 ) ⊕ O(−a2 ),
and −K · l = 2 − a1 − a2 > 0. Compare [D].
2
Proof of Lemma 2.50
Recall that we are assuming that −KW is very ample, that W contains a
surfaces S = Veronese surface or quartic scroll, and that the following holds:
|−KV | 63 D1 + D2 + nS,
with Di > 0 and S 6⊂ Di .
(irred mod S)
Write |D| = |−KW − S|.
Step 1 dim |D| ≥ 1.
Proof W = W2g−2 ⊂ Pg+1 is not a complete intersection, since S does not
generate Pic W ; hence by [Isk, Part I, 1.3], g ≥ 6. But hSi = Pn with n ≥ 5,
so that hyperplanes through S form at least a pencil.
Step 2 |D| is free and irreducible.
Let C ⊂ X ⊂ W be general curve and K3 sections. Since S and S ∩ X
are irreducible, H 1 (OW (−S)) = H 1 (OX (−S)) = 0, so that
H 0 (OW (D)) H 0 (OX (D)) H 0 (OC (D)).
By (irred mod S), the only possible fixed component of |D| is S itself. But
by the Exercise in Section 1, Step 1, or [Sh, 3.23], S can only be fixed if
4 = deg S = g − 1, which is absurd. Thus |D| has no fixed components, and
by arguing on the hyperplane section X, it is free; it is irreducible by (irred
mod S).
Step 3 Set |Dr | = |−KW − rS|, so that D = D1 ; if |Dr | is free for some
r ≥ 1 then dim ϕDr (W ) = 3.
11
Proof OS (−KW −rS) = OS (−KS −(r −1)S), so that the restriction of |Dr |
to S is a free linear subsystem of a very ample linear system on S. If ϕDr (W )
is a surface, it is necessarily this embedding of S, but then |Dr | decomposes
as a sum of mobile divisors; this contradicts (irred mod S).
Plan of proof
Before proceeding, I now outline the proof of Lemma 2.50 :
X ∈ |−KW | = |H| and Y ∈ |−KW − S| = |D|
will be nonsingular surfaces. In Step 4, I will show that dim |E| ≥ 2, where
|E| = |−KW − 2S|. Note that OY (KY ) = OY (−S), and that OY (E) =
OY (DY + KY ), where DY is the divisor class on Y corresponding to OY (D).
Furthermore, because H 1 (OW (−S)) = 0,
H 0 (OW (E)) H 0 (OY (E));
thus the fact that |E| is free at every point of Y will follow by the standard
Bombieri–Ramanujam method once we know that every element of |DY | is
numerically 2-connected. This occupies Step 5. Finally, we have to show that
a nonsingular Y ∈ |D| can be found passing through every P ∈ W .
Step 4 ϕD is birational and
(
16 if S = Veronese surface
g≥
15 if S = scroll F0;2 .
Proof Since dim hSi ≤ 5, H 0 (OW (D)) ≥ g − 4; thus W = ϕD (W ) is a
3-fold spanning Pn , with n ≥ g − 5, and hence deg W ≥ g − 7.
Write H = −KV ; from the multiplication table
(
1
H = 3 = 2g − 2; H 2 S = 4; HS 2 = 2; S 3 =
0
we get
(
2g − 21 if S = Veronese surface
deg ϕD deg W = D3 = (H − S)3 =
2g − 20 if S = scroll F0;2 .
Thus obviously ϕD = 1 and D3 ≥ g − 7 gives G ≥ 14 or 13 in the two cases.
We will now exclude deg W = g −7 and g −6. This increases the preceding
bound by 2, and proves the assertion. If deg W = g − 7 then W mus be a
12
scroll whose hyperplane section decomposes as a sum involving ≥ 2 mobile
divisors, contradicting (irred mod S). Thus now g ≥ 14 in either case.
If deg W = g − 6 with W spanning Pg−5 , and if P ∈ Sing W , then the projection of W from P is a scroll of degree ≥ 6, so that again W has hyperplane
sections decomposing as a sum of ≥ 2 mobile divisors, contradicting (irred
mod S). Thus W is nonsingular; since H 0 (OW (1)) is complete, it is easy to
check that the same is true of the hyperplane sections of W , and hence that
W is a Fano 3-fold of index 2. According to [Isk, Part I, 4.2, (ii)], these only
exist if deg W ≤ 7; this contradicts the above bound. Q.E.D.
Corollary dim |E| ≥ 1.
Proof ϕD (W ) = W spans Pn , with n ≥ g −4. On the other hand, OS (D) =
OS (−KS ), so that
(
9 if S = Veronese surface,
dim hϕD (S)i ≤
8 if S = scroll F0;2 ;
it follows that hyperplanes through ϕD (S) form at least a pencil. Q.E.D.
Step 5 By (irred mod S), the fixed part of |E| can be at worst a multiple
of S, so that |E − kS| = |−KW − (k + 2)S| has no fixed components for some
k ≥ 0. Now choose Y ∈ |D| to be nonsingular, and not to contain any of the
base curves of |E − kS|.
Claim OW (E) is generated by its global sections H 0 (OW (E)) at every point
P ∈Y.
Proof As above,
H 0 (OW (E)) H 0 (OY (E)) = H 0 (OY (DY + KY )).
Now by Bombieri [B, ??], to check that OY (D + KY ) is generated by its
sections at P ∈ Y , it is sufficient to have
(i) h0 (OY (D)) ≥ 3; (this is obvious).
(ii) Every divisor in |D| is numerically 2-connected.
I will drop the suffix Y on the divisor classes HY , SY , etc. duirng the rest
of this step, which is devoted exclusively to proving (ii).
Note first the following:
(1) H is very ample on V ;
13
(2) |D| = |H − S| is free, and D2 ≥ 8 (since ϕD is birational and because
of the bounds in Step 4);
(3) |H − (k + 2)S| has no fixed components for some k ≥ 0 (this follows by
choice of Y );
(4) KY = −S;
(5) S 3 = −3, HS = 3.
Now on any surface Y , |D| is free and D2 > 0 always implies that D is at
least 1-connnected: if D = E1 + E2 with Ei > 0 and E1 E2 ≤ 0, then DEi ≥ 0
gives
E12 ≥ −E1 E2 ≥ 0
E22 ≥ −E1 E2 ≥ 0.
But by the index theorem,
2
E1 E1 E2 ≤ 0,
det E1 E2 E22 with equality only if E1 ≡ qE2 (with q ∈ Q, q > 0; here ≡ denotes numerical
equivalence). This would contradict D2 > 0.
Now suppose that by some evil chance E1 E2 = 1, and (say) E12 ≤ E22 ; then
by the index theorem, and D2 ≥ 5, we get E12 ≤ 0. Since DE1 = E12 +E1 E2 ≥
0, there are 2 cases:
either: E12 = −1, DE1 = 0; this is impossible, because HE1 > 0 and
(H − S)E1 = 0 implies (H − (k + 2)S)E1 < 0, contradicting (3) above,
or: E12 = 0, DE1 = 1; then
(H − S)E1 = 1 and (H − (k + 2)S)E1 ≥ 0
give
(k + 1)SE1 ≤ 1 and HE1 = SE1 + 1 ≤ 2.
Now if S ≤ E1 , HE1 = H(E1 − S) + HS ≥ 3; hence S 6⊂ E1 , and so
SE1 ≥ 0.
Now because E12 = 0, KY · E1 = −SE1 ≡ 0 mod 2, and so necessarily
SE1 = 0; this gives HE1 = 1, and because H is very ample, E1 must be
an irreducible line. On the other hand, 2pa (E1 ) − 2 = E12 + KY E1 = 0,
which is absurd. This proves that every D ∈ |D| is 2-connected.
14
Step 6 |D| = |−KW − S|, |E|, etc., now revert to being linear systems on
W ; let DX , EX and so on denote their restrictions to X. If X ∈ |−KW | then
as usual H 0 (OW (D)) H 0 (OX (D)).
Now we know from Step 5 that |E| has no fixed components, so that the
complete linear system |EX | is free.2 On the other hand, |EX + 2SX | = |HX |
is ample, and it follows that the free system |DX | = |HX − SX | is also ample
(if Γ ⊂ X has HX · Γ > 0, DX · Γ = 0 then Ex · Γ < 0, contradicting the fact
that |EX | is free).
From Step 4 we know that ϕD |X = ϕDX is birational on X, and it follows
from [S-D, §6, p. 623] that |DX | is very ample on X; it is elementary to see
that |D| is then very ample on W . We can then choose a nonsingular Y ∈ |D|
through any point P , so that |E| is free on W . This completes the proof of
Lemma 2.50 .
3
Proof of Lemma 2.3–2.4
I assume throughout that V = V2g−2 ⊂ Pg+1 is a 3-fold embedded by −KV ,
and is an intersection of quadrics. I can assume that g ≥ 6, since the lines
and conics on the complete intersections are well understood.
Fix the following assumptions and notation:
(A) P ∈ V is a point not on any line; write σ : W → v for the blowup
of P ∈ V . Then S = σ −1 P = Veronese surface, −KW = σ ∗ (−KV ) − 2S,
(−KW )3 = 2g 0 − 2, where g 0 = g − 4 ≥ 2.
(B) q ⊂ V is a conic with NV |q ∼
= OP1 ⊕ OP1 or OP1 (1) ⊕ OP1 (−1); it
follows [Isk, Part II, 4.3] that q is not contained in a plane or quadric surface
of V . Write σ : W → V for the blowup of q ⊂ V . Then S = σ −1 q = quartic
scroll F0;2 or F2;1 , −KW = σ ∗ (−KV ) − S, (−KW )3 = 2g 0 − 2 where g 0 = g − 3.
2
Addendum, dating from 1982. The argument given here is over-hasty. The following is
more complete: since S is the only possible fixed component of |E|, and S ∩ Y = SY 6= 0,
we know from Step 5 that |E| has no fixed components on W . I thus choose X ∈ |−KW |
such that
(a) P ∈ X;
(b) X is nonsingular;
(c) X does not contain any base curve of |E|, and
(d) ϕ−KW −S |X is birational.
Now |EX | ⊃ |E|X has no fixed components, and because |EX + 2SX | is ample, it follows
that also |EX +SX | is ample (because for Γ ⊂ X, 2(EX +SX )·Γ = EX Γ+(EX +2SX )Γ > 0).
But then because of (d) and [S-D, §6, p. 623], DX is very ample on X, and we can find
Y ∈ |D| such that Y is nonsingular and passes through P . This shows that |E| is free, and
completes the proof of Lemma 2.50 .
15
For C ⊂ W a curve not in S, write σ(C) = Γ ⊂ V .
Lemma 2.3–2.4 will be a consequence of the following more precise result of Fano–Iskovskikh type; compare [Isk, Part II, §5]. The deduction of
Lemma 2.3–2.4 requires some slightly tricky arguments in the cases g = 6
or 7 (Steps 6 ffoll. below).
Theorem |−KW | is free and defines a generically finite morphism ϕ−KW ;
π
write ϕ : W → W for the Stein factorisation, ϕ : W → W −→ ϕ−KW (W ) ⊂
0
Pg +1 . Then ϕ is a small birational morphism, W is Gorenstein with ample
−KW and ϕ∗ (−KW ) = −KW . We have H i (OW ) = 0 for i = 1, 2 and 3.
Suppose in addition that |−KV | 63 D1 +D2 with Di > 0. Then for C ⊂ W
a curve not in S and with C · (−KW ) = 0, we have
(A) if g ≥ 8 then Γ is a conic through P ;
(B) if g ≥ 7 then Γ is a line meeting q.
In cases g ≤ 7 we may also have a finite number of curves C ⊂ W of the
following types:
(A) g ≤ 7: C ∼
= P1 , with deg Γ = 4, multP Γ = 2, hΓi = P3 , pa Γ = 1.
g = 6: C ∼
= P1 , with deg Γ = 6, multP Γ = 3, hΓi = P4 , pa Γ = 2.
(B) g = 6: Γ is a conic such that hΓ ∪ qi = P3 , pa (Γ ∪ q) = 1.
Remark 1 there are only finitely many conics through P if g ≥ 9 by [Isk,
Part II, 4.4, (iii)], and similarly, finitely many lines meeting q if g ≥ 8.
Remark 2 The fact that general projections contract finitely many rational
curves to double points occurs throughout Fano’s work. In special cases one is
allowed to assume that NW |C ∼
= OP1 (−1) ⊕ OP1 (−1) for the contracted curves
C, which ensures that W has ordinary double points (xy = zt); singularities
obtained by contracting curves C with NW |C ∼
= OP1 ⊕ OP1 (−2) include xy =
z 2 + t2n for all n ≥ 2. The significance of these singularities in providing
elementary transformations in the birational geometry of 3-folds is beginning
to be appreciated.
Step 1 |−KW | is free and ϕ−KW is generically finite; the general element
X ∈ |−KW | is a K3 surface, and H 0 (OW (−KW )) H 0 (OX (−KW )). There
is no surface F ⊂ W for which ϕ−KW (F ) = pt.
Proof I consider (A) only. (B) requires only trivial modifications.
Write TP V = P3 for the tangent space to V at P ; then ϕ−KW will be
interpreted as the linear projection of V from TP V . We have σ∗ OW (−KW ) =
m2P · OV (−KV ), where mP is the maximal ideal of P , so that
|−KW | = σ ∗ div(s) − 2S s ∈ H 0 (m2P · OV (−KV )) .
16
By definition of the blowup mrP · OW = OW (−rS) for r ≥ 1, so that |−KW |
is free3 if and only if m2P · OV (−KV ) is generated by H 0 (m2P · OV (−KV ));
geometrically, this is the assertion that TP ∩ V = Z, where Z ⊂ V is the
subscheme defined by m2P ; this in turn follows from the fact that V is an
intersection of quadrics, and that P is not on any line: indeed, TP V ∩ V is an
intersection in TP V = P3 of quadrics singular at P . This proves that |−KW |
is free; (−KW )3 > 0, so that ϕ−KW has 3-dimensional image. If F ⊂ W is
a surface contracted to a point by ϕ−KW , F 6= s, then hσ(F ) ∪ TP V i = P4 ;
but then σ(F ) ∩ Tp V would have dimension ≥ 1, contradicting TP V ∩ V = Z.
The remaining assertions are standard.
Step 2 I now discuss −KW .
By construction of ϕ, W is normal, and OW (−KW ) = ϕ∗ OW (1) for some
0
0
'
ample sheaf OW (1) on W . Now write ϕ : W 0 −→ W for an open set W ⊂ W
0
with codim(W \ W ) ≥ 2; then Ω3W ∼
= Ω3W when restricted to this open.
But if follows from the description of OW (KW ) as a divisorial sheaf (see [R,
Appendix to §1] that OW (KW ) ∼
= OW (−1).
∗
∼
Now since ϕ OW (KW ) = OW (KW ), it follows from the projection formula
and the vanishing Ri ϕ∗ OW (KW ) = 0 of [G–R] that Ri ϕ∗ OW = 0 for i = 1, 2
and 3. Thus W has rational singularities, and is hence Gorenstein. H i (OW ) =
0 follows from the Leray spectral sequence.
Step 3 Let C ⊂ W be any curve not in S and contracted by ϕ−KW , and
suppose CS ≥ 2. Then the intersection C ∩ S is part of the singular locus of
ϕ−KW |S .
Proof Z = C ∩ S is a subscheme of S with deg Z = length(OZ ) = CS ≥ 2,
and such that
H 0 (OW (−KW )) → H 0 (OZ (−KW ))
3
Erratum dating from 1982: G. Horrocks has kindly pointed out that this section contains two errors, that fortunately cancel out. A correct argument is as follows:
From the assumption that V is an intersection of quadrics, and that P is not on any line
of V , we see that TP V ∩ V = {P }; indeed, this is an intersection if TP V = P3 of quadrics
singular at P . Now in affine coordinates at P ∈ Pg+1 , and quadric Q with V ⊂ Pg+1 .
Now in affine coordinates at P ∈ Pg+1 , any quadic Q with V ⊂ Q ⊂ Pg+1 has
equation q = q1 + q2 with qi homogeneous of degree i; q1 defines TP Q, and because
TP V = bigcapTP Q, these q1 give the whole of H 0 (m2P · OV (−KV )).
Choosing a local generator of OV (−KV ), and s ∈ H 0 (m2P · OV (−KV )) can be considered
as an element of m2P , with leading term s2 = s ∈ m2 /m3 . Obviously |−KW | is free if and
only if the s2 define a collection of quadric cones in TP V = A3 whose intersection if {P },
correspodning to the empty set in P(TP V ) = S. But this is clear: if s = q1 then s2 = q2 ,
and the s2 define TP V ∩ V = {q1 = q2 = 0 for all V ⊂ Q ⊂ Pg+1 } = {P }
17
has rank 1. Q.E.D.
Step 4 Assume that |−KV | 63 D1 + D2 with Di > 0. Then the restriction
ρ : H 0 (OW (−KW )) → H 0 (OS (−KW )) has maximal rank.4
Proof I suppose that ρ is not injective, and D ∈ |−KW − S|; it is enough
to show that H 1 (OW (D)) = 0. By the hypothesis on |−KV |, D = rS + E for
some r ≥ 0, with E 6= 0 irreducible.
Now let X ∈ |−KW | be a K3 surface meeting E and S in irreducible
curves EX , SX . The exact commutative diagram
k
T
=
H 0 (OW (−KW )) →
k
T
H 0 (OS (−KW ))
↓
→
H 1 (OW (D))
→ 0
↓
H 0 (OX (−KW )) → H 0 (OSX (−KW )) → H 1 (OX (DX )) → 0
shows that H 1 (OW (D)) ∼
= H 1 (OX (DX )). But DX is obviously numerically
connected:
2
DX = EX + rSX , HX · SX = 4, SX
= −2,
so that for b ≥ 1,
(EX + (r − b)SX )bSX = b(HX − (b + 1)SX )SX > 0.
Thus h0 (ODX ) = 1, H 1 (OX (−DX )) = 0, and by Serre duality we have
H 1 (OX (DX )) = 0. Q.E.D.
Step 5 To prove the theorem, I now analyse the curves C ⊂ W with
C · (−KW ) = 0; I start with case (A).
If g ≥ 8, then h0 (OW (−KW )) = g − 2 ≥ 6 = h0 (OS (−KW )). By Step 4,
ϕ−KW embeds S in P5 as the Veronese surface. According to Step 3 the only
curves C ⊂ W contracted by ϕ−KW have CS ≤ 1. Then
0 = C · (−KW ) = C · (σ ∗ (−KV ) − 2S) = deg Γ − 2CS,
(†)
where CS = multP Γ; this is only possible for Γ a conic through P .
Now if C is a contracted curve we have deg Γ = 2 multP Γ from (†); and
from the description of ϕ−KW in terms of projections of V from TP V , we must
have hΓ ∪ TP V i = P4 .
Now V ∩ P4 is an intersection of quadrics, and all its components are of
dimension ≤ 1. There are only the following possibilities for dim Γ = 1:
4
Mori says that this part is wrong.
18
hΓi = P2 , Γ a conic;
hΓi = P3 , deg Γ = 4, Γ an intersection of 2 quadrics;
hΓi = P4 , deg Γ = 6, Γ a component of an intersection of quadrics;
hΓi = P4 , deg Γ = 8, and Γ the complete intersection of 3 quadrics.
The final case is impossible; for on the one hand by the adjunction formula
in P4 , ωΓ = OΓ (1); but on the other hand, Γ is a component of a scheme Z
which is a curve section of V , and also has ωZ = OZ (1). Since g ≥ 6,
deg Z = 2g − 2 > 8, and Z has another component meeting Γ. This is
impossible.
It is easy to see that in each case the blowup C of P ∈ Γ is rational and
nonsingular; the fact that any curve Γ with hΓi = P4 , deg Γ = 6, multP Γ = 3
has pa Γ = 2 will not be used, and is left as a treat for the reader.
The final assertion is the finiteness of the number of contracted curves.
For this we use Step 3, and the geometry of ϕ−KW |S : S → S # ⊂ Pg−3 ; this
is the famous theory of projections of the Veronese [S–R, p. 128]. If g = 7
then either S → S # ⊂ P4 is an embedding, or S # has a line of singularities,
corresponding to a projection from a vertex lying in the plane of some conic.
If g = 6 then S → S # ⊂ P3 is the Steiner surface, with 3 double lines meeting
in a triple point.
By Step 3, any of the exceptional contracted curves map to singularities
of S # ; the fact that there are now only finitely many of these now follows: if
F ⊂ W is a surface contracting to a line under ϕ−KW then |−KW − F | =
6 0
contradicting |−KV | 63 D1 + D2 with Di > 0.
(B) is exactly similar; for the final finiteness assertion I need to know that
a 1-dimensional component of the singular locus of ϕ−KW : S → S # ⊂ P4 is
a line. This follows for example by arguing on the arithmetic genus of the
general section of S # .
This complete the proof of the theorem of this section. I now embark on
the proof of Lemma 2.3–2.4.
Step 6 If ϕ−KW is finite then W = W , and −KW is ample; W cannot be a
hyperelliptic Fano 3-fold, since then by [Isk, Part I, 4.2] we would have either
Pic W ∼
= Z or |−KW | 3 D1 + D2 with Di > 0.
In (A) when g ≥ 8, the obstruction to the finiteness of ϕ−KW consists of
the conics through P ; in (B) when g ≥ 7, it consists of the lines meeting q.
This proves Lemma 2.3–2.4 in these cases.
I now outline the proof of Lemma 2.3–2.4 in the cases g = 6 or 7. In the
diagram
we have the following information:
19
(i) There are finitely many curves Ci ⊂ W with ϕ(Ci ) = Pi ∈ W , and ϕ is
'
an isomorphism W \ {Ci } −→ W \ {Pi }. Every point Pi is a singular
point of S and of W :
{Pi } = Sing S = Sing W .
(ii) ϕ−KW is either an embedding of W as a quartic in P4 or a finite double
cover of a nonsingular quadric Q ⊂ P4 or P3 ; it is easy to see that the
proof of [Isk] carries over to singular W once we know tht |−KW | is
free. The fact that the singular quadric cannot turn up follows from
the irreducibility assumption on |−KV |.
0
(iii) The composite π : S → S # ⊂ Pg +1 is the linear projection of S ⊂ P5 ,
where S is the Veronese surface or the scroll F0;2 .
After dividing into cases, I will in each case either prove that ϕ is finite, or
deduce a contradiction directly. It is convenient to state first the properties
of the projections to P4 of F0;2 ⊂ P5 which I will need. Recall that F0;2 is
P1 × P1 embedded by |2A + B|; a ∈ |A| is a line of the ruling, and b ∈ |B| a
conic.
Proposition Let S = F0;2 ⊂ P5 , and consider the projection π : S → S # ⊂
P4 from a point O ∈ P5 not in S. Then either (i) or (ii) below holds:
(i) O lies in the plane hbi = P2 of some conic b ∈ |B|; π : S → S # maps b
two-to-one onto a double line l ⊂ S # , l = Sing S # .
(ii) O lies on a unique secant or tangent line of S. The surface S # has one
double point P ∈ S # , and the projectivised tangent cone Z = P(TP S # )
is either a pair of skew lines, or a double line not lying in any plane P2 .
In particular Z is not contained in any quadric of rank 3.
Proof Every O ∈ P5 lies on at least one secant line l of S; l ∩ S contains a
finite scheme L ⊂ l ∩ S of degree 2 (in fact L = l ∩ S), and L is not contained
in any a ∈ |A|, since then l = a ⊂ S. There are thus two cases (see Figure 1)
(i) L ⊂ b for some b ∈ |B|;
Figure 1: (i) L ⊂ b or (ii) L 6⊂ b.
20
(ii) L 6⊂ b.
There is just one of each of these cases up to projective equivalent, and it is
elementary to give the parametrisation P1 × P1 → S # and to verify the above
assertions in each case. The final exotic case goes as follows: in coordinates
(x, y) × (s, t) on P1 × P1 , let L be the tangent vector at P = (1, 0) × (1, 0)
to ys = xt. The parametrisation of S # is (xs2 , xst − ys2 , xt2 , yst, yt2 ). In
inhomogeneous coordinates at P this can be written (1, t − y, t2 , yt, yt2 ), and
the equations of S # can be written
u1 u2 u3 u4 u2 u21 u4 u1 u3 = 0
2
(that is, set to zero the 2 × 2 minors), where
u1 = t2 ,
u2 = t3 ,
u3 = t − y,
u4 = t2 − ty.
The tangent cone Z is given by
u 1 u4 = u2 u3
and u22 = u2 u4 = u24 = 0,
which is a (2, 0)-divisor on the nonsingular quadric Q ⊂ P3 given by u1 u4 =
u2 u3 (see Figure 2). Hence Z is not contained in a plane. Finally, every line
Figure 2: The subscheme Z in the quadric Q : u1 u4 = u2 u3 .
pair or double line in a quadric of rank 3 of P3 is contained in a plane. Q.E.D.
0
Step 7 I now deduce a contradiction assuming that ϕ−KW : W → Pg +1 is
a finite double cover of P3 or of a nonsingular quadric Q ⊂ P4 .
In either case S # is a complete intersection, and necessarily has a double
line l. I show that the double cover splits over l; this contradicts the fact that
in S → S # the double line unzips into an irreducible conic.
Note that W , and hence the branch locus B, has only finitely many singularities; B is thus irreducible, and does not contain S # . Since S → S #
has degree 1, the double cover splits over S # , and so B lifts back to twice
an effective divisor on S. B cannot contain l, because then it would have to
contain the tangent cone to S # at a general point of l, and thus be singular
along l. Hence B touches l.
21
Step 8 I now assume that ϕ−KW : W ,→ P4 .
In this case S = S # , so that S # has only finitely many singularities. In
case (A), S # is then necessarily the nonsingular projection of the Veronese,
'
so that by Step 3 ϕ : W −→ W , and we are home.
There remains to eliminate the case that S # is one of the projections
of F0;2 of (ii) of the Proposition in Step 6. I will show (following an idea
suggested by Fabrizio Catanese) that in this case the only singularity of W is
an ordinary double point. We will then get a contradiction by the next easy
result.
Lemma 2.50 Let X ⊂ P4 be a hypersurface of degree ≥ 3 having an ordinary
double point P ∈ X as its only singularity. Then every surface S ⊂ X is the
intersection of X with a hypersurface.
Proof Let σ : F → P4 be the blowup of P ∈ Sing X; Then σ ∗ X = σ 0 X +2B,
where B is the exceptional component, and σ 0 X is a smooth very ample
divisor on F . By the Lefschetz theorem, Pic σ 0 X ∼
= Pic F . Every irreducible
0
0
divisor on s X is then the intersection of σ X with an irreducible divisor on
F in the class B of σ ∗ (aH − bB) with a ≥ b, as follows from the fact that
H 1 (F, L) for all L ∈ Pic F .
The singularity P ∈ W is also a singular point of S, and the tangent
cont to W at P contains the tanget cone to S at P ; thus it is certainly not
a quadric of rank 3! It will therefore be enouth to prove that the general
hyperplane section through P ∈ W is a surface X having an ordinary double
point.
Claim Let q and Γ be as in the exception case for (B), g = 6 in the theorem,
that is, hΓ ∪ qi = P3 , pa (Γ ∪ q) = 1. then the general hyperplane section
X ∈ |−KV | passing through q ∪ Γ is nonsingular.
This will clearly do what we want. When we blow up Q, σ : W → V , we
get σ ∗ X = S + σ 0 X, where σ 0 X is isomorphic to X, and maps to X under
ϕ; the singularity of X can thus be resolved by introducing just one curve
C∼
= P1 , and this is well known to characterise ordinary double points.
The claim is proved by an easy dimension count: hyperplanes of Pg+1
containing hΓ ∪ qi = P3 form an ∞g−3 , and one checks that the hyperplanes
containing the tangent space to V at one of the sngularities of Γ∪q,a nd those
containing the tangent space to V at a general point of Γ or q are ∞g−4 .
This complete the proof of Lemma 2.3–2.4.
Remark There is another beautiful idea in [Sh]: if V is a Fano 3-fold, then
the splitting of the curve sections of V also has strong consequences for V .
22
For example, if g ≥ 5 and a curve section splits as Γ ∪ Γ0 with deg Γ = 2 or
3, and pa Γ = 1, then V is necessarily hyperelliptic or trigonal.
Another example: if g = 6 then curve sections of V10 ⊂ P7 might split as
Γ ∪ Γ0 , where
deg Γ = 4, hΓi = P3 , pa Γ = 1
deg Γ0 = 6, hΓi = P4 , pa Γ = 2.
Now if there exists one such splitting there will exist many – in fact Γ will
move in an elliptic pencil, and Γ0 in a 2-dimensional linear system on every
K3 section of V . This is a proerty that V inherits from the geometric form
of Riemann–Roch on its general curve section.
Shokurov uses this idea [Sh, 6.15 and 7.8] to show that if the contracted
curves Γ in the theorem of this section exist, then either they meet lines or
conics of V , or the projection V 99K P3 from a general such curve will be a
finite double cover, leading to a contradiction as usual.
One might be tempted to think (in analogy with [R2]) that then necessarily V has a pencil of surfaces cutting out Γ on its K3 sections. This is not
the case, since one can obviously construct a nonsingular V = Q ∩ H1 ∩ H2 ∩
Grass(2, 5) having Pic V = Z, and mpose some elliptic curve of degree 4 in
Grass(2, 5) on V . This incidentally shows that there exist V with Pic V = Z,
but for which every K3 section X has rank Pic V ≥ 2. I have some reason
for believing that this happens for every V in Iskovskikh’s terminal family of
Fano 3-folds with g = 12 [Isk, Part II, §6].
References
[Sh]
V.V. Shokurov, The existence of lines on Fano 3-folds, Izv. Akad. Nauk
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[Sh2] V.V. Shokurov, The smoothness of the general anticanonical divisor on
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[Isk2] V.A. Iskovskikh, Anticanonical models of algebraic 3-folds, Contemporary problems in Math, Itogi nauki i tekhniki. Translated in J. Soviet
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[S-D] B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math 96
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V.G. Sarkisov, Birational automorphisms of conic bundles, Izv. Akad.
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V.G. Sarkisov, On the structure of conic bundles, Izv. AN SSSR, Ser.
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I. V. Dëmin, Fano threefolds that can be represented as P1 -bundles
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S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133–176
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M. Reid, Canonical 3-folds, in Journées de géométrie algébrique d’Angers, A. Beauville (ed.), Sijthoff and Noordhoff, Alphen (1980) 273–310
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Henry B. Laufer, On CP1 as an exceptional set, in Recent developments
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24