Algebraic Geometry and Commutative Algebrain Honor of Masayoshi NAG ATA
pp. 357-377 (1987)

Curves, K3 Surfaces and Fano 3-folds of Genus < 10
Shigeru Mukai"
A pair (5, L) of a K3 surface S and a pseudo-ample line bundle L on 5 with(L^) = 2p - 2 is called a (polarized) K3 surface of genus g. Over the complex
number field, the moduli space Tg of those (5, L)'s is irreducible by the Torelli
type theorem for K3 surfaces [12]. If L is very ample, the image S2g-2 of #|£,|
is a surface of degree 2p — 2 in P® and called the projective model of (5,1/),[13]. If p = 3,4,5 and (5, L) is general, then the projective model is a complete

intersection of p - 2 hypersurfaces in P^. This fact enables us to give an explicit
description of the birational type of J^g for g <5. But the projective model is no
more complete intersection in P® when g >6. In this article, we shall show thata general K3 surface of genus 6 < ^ < 10 is still a complete intersection in acertain homogeneous space and apply this to the discription of birational type ofJ^g for 5 < 10 and the study of curves and Fano 3-folds. The homogeneous space
X is the quotient of a simply connected semi-simple complex Lie group G by ameiximal parabolic subgroup P. For the positive generator Ox(l) of PicX = Z,

the natural map X -* P{H°{X,Ox{l)) is a G-equivariant embedding and theimage coincides with the G-orbit G • u, where u is a highest weight vector of theirreducible representation ir®(X,Gjc(l))^ of G. For each 6 < ^ < 10, G and
the representation U = H^{X,Ox{l)) are given as follows:
710

Spm(lO)

exceptionalof type Gj

d i m G

45

35

U

half spinor

A2y0

21

14

representationadjoint
representationd i m G

16

14

d i m X

10

5

where V denotes an t-dimensional vector space and <r € is a non-degenerate 2-vector of F®.
*Partically supported by SPB 40 Theoretische Mathematik at Bonn and Educational
Projects for Japanese Mathematical Scientists.
Received April 7, 1987.

358

S. MukaiIn the case 7 < 5 < 10, dim U is equal to p + n - 1, n = dimX. X is ofdegree 2p - 2 in P{U) ^ P^'+"-2 and the anticanonical (or 1st Chern) classof X is n - 2 times hyperplane section (cf. (1.5)). Hence a smooth complete
intersection of X = X2g-2 and n - 2 hyperplanes is a K3 surface of genus g.
(This has been known classically in the case g = 8 and is first observed by C.
Borcea [1] in the case g = 10.)

Theorem 0.2. If two K3 surfaces S and S' are intersections of X2g-2 (7 <g < 10) and g-dimensional linear subspaces P andP', respectively, and if S C P
and S' C P' are projectively equivalent, then P and P' are equivalent under theaction of G on P{U), where 0 is the quotient of G by its center.

By the theorem there exists a nonempty open subset S of the Grassmannvariety G{n - 2,17) of n - 2 dimensional subspaces of U such that the naturalmorphism S/G Pg\s injective. For each 7 < p < 10, it is easily checked that
dimS/G = 19 = dhaTg. Hence the morphism is birational.

Corollary 0.3. The generic K3 surface of genus 7 < g < 10 is a complete
intersection of X2g-2 C P(17) and a g-dimensional linear subspace in a unique
way up to the action G on P{U). In paHicular, the moduli space Tg is birationally equivalent to the orbit space G{n — 2, U)fG.
In the case 5 = 6, the generic K3 surface is a complete intersection of X,

a linear subspace of dimension 6 and a quadratic hypersurface in P{U) = P®.

We have a similar result on the uniqueness of this expression of the K3 surface(see (4.1)). In the proof of these results, special vector bundles, instead of line
bundles in the case g < 5, play an essential role. For instance, the generic K3

surface (5, L) of genus 10 has a unique (up to isomorphism) stable rank two
vector bundle with ci{E) = ci{L) and C2{E) = 6 on it and the embedding of S
into X = GIP is uniquely determined by this vector bundle E.

The following is the table of the birational type of 7^^ for g < 10:
(0.4)

^

genus

234

birational type I P(S"£/^)/PGL(3) P(S''C7'»)/PGL(4) P(17'*")/SO(5)5

6

7

G(3,5='G»)/PGL(6) (C7" ® C7")/PGL(2) G(8,f7^«)/PSO(10)10

G(6, A='F«)/PGL(6) G(4, C7")/PSp(3)G(3,fl)/G2

K3 Surfaces and Fano 3-folds

359

where is a d-dimensional irreducible representation of the universal coveringgroup.

Corollary 0.5. is unirational for every g < 10.

By [5], there exists a Fano 3-fold V with the property PicV = Z(—iiTv) and= 22. The moduli space of these Fano 3-folds are unirational by theirdescription in [5]. The generic K3 surface of genus 12 is an anticanonical divisorof V and hence .F12 is also unirational.
_ Problem 0.6. Describe the birational types, e.g., the Kodaira dimensions, ofthe 19-dimensional varieties for 5 > 0. Are they of general type?
If {S,L) is a K3 surface of genus g, then every smooth member of jLj is a
curve of genus g. Conversely if C is a smooth curve of genus 5 > 2 on a K3
surface, then Os{C) is pseudo-ample and (5, Cs(C)) is a K3 surface of the same
genus as C. In the case g <9, the generic curve lies on a K3 surface, that is ,the natural rational map

4>g ■'Pg = U 1^1 ' "^3 ~ moduli space of curves of genus g)( s , L ) e : F,is generically surjective (§6). The inequality dim Mg < ddmVg = 19+ g holds
if and only if p < 11 and ipu is generically surjective ([10]). But in spite ofdim Mxa =27 < dim Via = 29, we have

Theorem 0.7. The generic curve of genus 10 cannot lie on a K3 surface.

Proof. Let (resp. A^io) be the subset of ^10 (resp. Afio) consisting ofK3 surfaces (resp. curves) of genus 10 obtained as a complete intersection in

the homogeneous space Xfg C P(fl). M'xq has a dominant morphism from a
Zariski open subset U of (?(4, fl)/G. Since the automorphism of a curve of genus> 2 is finite, the stabilizer group is finite for every 4-dimensional subspace of 0

which gives a smooth curve of genus 10. Hence we have dim At'jo ^ dim U =dim G(4, fl) - dim G = 26 < dim Miq. On the other hand contains a dense
open subset of ^10 by Theorem 0.2. Hence the image of ^10 is contained in theclosure of = Vio(-^'io) and Vio is not generically surjective. q.e.d.
Remark 0.8. Every curve of genus 10 has a 4-dimensional linear system ofdegree 12. If C is a general linear section of the homogeneous space Xig C P*®,

then every pfj ^ embeds C into a quadric hjrpersurface in P^. But if C is
the generic curve of genus 10, then the image C12 C P^ embedded by any

not contained in emy quadratic hypersurface. This fact gives an alternate proofof the theorem.

In the case 7 < 5 < 10, a Fano 3-fold V2g-2 C is obtained as acomplete intersection of the homogeneous space and a linear subspaceof codimension n — 3 in P{U) = P"+3-2 gy Lefschetz theorem, the Fano
3-fold V = V2g-2 has the property PicV = Z{—Kv)- The existence of such V

has been known classically but was shown by totally different construction {[6]).
Theorem 0.2 holds for Fano 3-folds, too.
Theorem 0.9. Let V2g-2 and (7 < 5 < 10) be two Fano 3-folds which
are complete intersections of the homogeneous space C and linear

subspaces of codimension n-3. IfV2g-2 and Vfg_2 are isomorphic to each other,
then they are equivalent under the action of G.

We note that, by [1], the families of Fano 3-folds in the theorem is locallycomplete in the sense of [7].
The original version of this article was written during the author's stay at
the Max-Planck Institiit fiir Mathematik in Bonn 1982 and at the Mathematics

Institute in University of Warwick 1982-3. He expresses his heeirty thanks toboth institutions for their hospitality. He also thanks Mrs. Kozaki for nicetyping into I^TgX.
Conventions. Varieties and vector spaces are considered over the complex

number field 0. For a vector space or a vector bundle E, its dual is denoted by
For a vector space V, G{r,V) (resp. G(V, r)) is the Grassmann variety ofr-dimensional subspaces {resp. quotient spaces) of V. G{l,V) and G(V, 1) aredenoted by P»(V) and P{V), respectively.
§1. Preliminary

We study some properties of the Cayley algebra C over C. C is an algebra
over C with a unit 1 and generated by 7 elements Cj, 1 € Z/7Z. The multipli
cation is given byQ
a n di ee i Z/7Z
C i + a and= -ae .—
+ a 1,2,4.
Ci ^ ' =for- 1every
The algebra C is not associative but alternative, i.e., x(xy) = x^y and{xy)y = xy^ hold for every x,y e C. Let Cq be the 7-dimensional subspace of C
generated by ei,i € Z/7Z and U the subspace of Cq spanned by a = 63 -f- ^/^ei

and p = Cq - y/^er. It is easily checked that = aP = Pa = 0, i.e., U
is totally isotropic with respect to the multiplication of C. Moreover, U is max
imally totally isotropic with respect to the multiplication of C, i.e., if xU = 0 orZ7a; = 0, then x belongs to U. Let q be the quadratic form q{x) = x^ on Cq andb the associated symmetric bilinear form. b{x, y) is equal to xy -t- yx for every x
and y £ Cq. Let V be the subspace of Cq of vectors orthogonal to U with respect
to q (or b). Since U is totally isotropic with respect to q, V contains U and thequotient VfU carries the quadratic form q.

K3 Surfaces and Fano 3-folds

361

Lemma 1.2. x'{xy) = b{x, y)x' — b{x', y)x + y{x'x) for every x, x' and y 6 Cq.
Proof. By the alternativity of C, we have u{vw) + v{uw) = {uv + vu)w.
Hence, if u and v belongs to Co, then we have u{vw) +v{uw) = b{u,v)w. So wehavex'{xy) = x'{b{x,y) - yx) = b{x,y)x'- x'{yx)= 6(x, y)x' - {b{x\y)x - y{x'x))= 6(x,y)x'- 6(x',y)x + y(x'x).
q.e.d.

If X € U and y 6 V, then U{xy) = 0 by the above lemma and hence xy
belongs to U. Hence the right multiplication homomorphism i2(y), x xy, byy €V maps U into itself. Since R{x) is zero on U if and only if x € 17, i? gives
an injective homomorphism R : V/U —» End(17).

Proposition 1.3. (1) R{x)^ = q{x) - id for every x e V/U, and(2) R is an isomorphism onto sl{U), the vector space consisting of trace zero
endomorphisms of U.

Proof. (1) follows immediately from the alternativity of C. It is easy tocheck the following fact: if r is an endomorphism of a 2-dimensional vector
space and if is a constant multiplication, then either r itself is a constantmultiplication or the trace of r is equal to zero. Hence by (1), R{x) is a con

stant multiplication or belongs to sl{U), for every x € V/U. Therefore, R{V/U)
is contciined in the 1-dimensional vector space consisting of constant multipli
cations of U or contained in the 3-dimensional vector space sl{U). Since thequadratic form q is nondegenerate on V/U, the former is impossible and R{V/U)coincides
withsl{U).
q.e.d.
Let G be the automorphism group of the Cayley algebra C. It is known that
G is a simple algebraic group of type G2. The automorphisms which map U
onto itself form a maximal parabolic subgroup P of G. The subspace spannedby ei, 62 and 64 (resp. by 63 - es and 67) can be identified with
3l{U) (resp. 17^) by R (resp. 6). C is isomorphic to C © 17 ® sl{U) © Z7^ and if/ e GL(Z7), then 1 © / ® ad(/) © '/ is an automorphism of the Cayley algebra C.
Hence the maximal parabolic subgroup P contains GL(17) and X = G/P canbe identified with the set of 2-dimensional subspaces of Cq which are equivalent
to U under the action of G = Aut C.

Let U be the maximally totally isotropic universal subbimdle of Co ® Ox'
the fibre Ux C Co at x is the 2-dimensional subspace corresponding to x e JX".
Let V be the subshe£if of Co ® Ox consisting of the germs of sections which areorthogonal to U with respect to the bilinear form 6 ® 1 on Co ® Ox' V is a rank

5 subbundle of Co ® Ox and contains U as a subbundle. The quotient bundle

362

S. Mukai(Co ® Ox)lV is isomorphic to by 6 ® 1 and VfU has a quadratic form 9 ® 1

induced by g ® 1 on Cq ® Ox- By Proposition 1.3, we have

Proposition 1.4. The right multiplication induces an isomorphism R from

V/U onto the vector bundle sl{U) of trace zero endomorphisms ofU and R{xyis equal to {q ® l)(z) • id for every x 6 V/U.

Next we shall compute the anticanonical class of X and the degree of Ox{\),the ample generator of PicX, and show some vanishings of the cohomology
groups of homogeneous vector bundles U{i) and (S^W)(t) etc.
Let G be a simply connected semi-simple algebraic group and P a maximal
parabolic subgroup of G. Fixing a Borel subgroup B in P, the Lie algebra g of
G is the direct sum of b and 1-dimensional eigenspaces where runs over all
negative roots. If we choose a suitable root basis A, then there exists a positiveroot a € A such that p is equal to the direct sum of 0 and b, where 7 runs
over all positive roots which are linear combinations of the roots in A \ {a}with nonnegative coefficients. A positive root /3 is said to be complementary 'i£g'' n p = 0 or equivalently if p cannot be expressed as a linear combination of
the roots in A \ {a} with nonnegative coefficients.

Proposition 1.5. (Borel-Hirzebruch [2]) Let G, P, A and a be as above and
L the positive generator o/Pic(G/P). Then we have(1) the quotient g/p is isomorphic to 9^, where Rp is the set of pos
itive complementary roots. In particular, dim(G/P) is equal to the cardi
nality n of Rp,

where w is the fundamental weight corresponding

to a {or L) and p is a half of the sum of all positive roots, and(3) the sum of all P € Rp is r times p for some positive integer r and ci (G/P)(or the anticanonical class of G/P) is equal to r times ci{L).
A homogeneous vector bundle on G/P is obtained from a representation of

P and hence from that of reductive part Go of P. Note that the weight spacesof G and Gq are naturally identified.

Theorem 1.6. (Bott [3]) Let E be a homogeneous vector bundle over G/P
induced by an irreducible representation of the reductive part of P. Let 7 be thehighest weight of the representation and p a half of the sum of all positive rootsof G. Then we have(1) ifilf + PtP) = 0 for a positive rootp, then H*{GlP,E) vanishes for every
i, and(2) let »o be the number of positive roots P with (7 4- p,P) negative (zq iscalled the index of E). Then H*{G/P,E) = 0 for all i except for t'o andH^°{GIP,E) is an irreducible G-module.

K3 Surfaces and Fano 3-folds

363

Returning to our first situation, our variety X is the quotient of the excep
tional Lie group G of type G2 by a maximal parabolic subgroup P. The rootsystem G2 has two root basis ai and 02 with different lengths Jind the root acorresponding to P in the above manner is the longer one, say 02- The line bun

dle L = Ox{l) and the vector bundle W^(l) on X come from the representation
with the highest weights wi = 3ai + 2a2 and W2 = 2ai + 02, respectively, which

are the fundamental weights of G. Since W is of rank 2 and A^U = Ox{l) >is isomorphic to U{1). p is equal to Wi -f W2 and the inner products of p, wi,W2
and the 6 positive roots are as follows:

By (1.5), X has dimension 5, Ci(X) = 3ci(L) and has degree, 3.3.6.3.3^'■ ' 6 - 5 - 9 - 4 - 3

in The homogeneous vector bundles (S"*W)(n) comes from the irreducible
representation with the highest weight mwi -1- (n — m)w2. Applying (1.6), we
have

Proposition 1.7. The cohomology groups o/W(n), (S^W)(n) and (S^W)(n) are
zero except for the following cases:(1) H\X,U{n)) for n > l,H\X,{S^U){n)) forn>2 and H'^{X,{S^U){n))for n > 3,

^ (2) H'{X,{S^U){1)) andH''{X,{S^U){-l)), and(3) J?'^(A:,W(n)),fl'5(X,(S2W)(n)) and H^{X,{S^U){n)) for n < -3.

Let 5 be a smooth K3 surface which is a complete intersection of 3 membersof 10^^(1)1. By using the Koszul complex(1.8) 0 —» C?x(-3) —^ C7A:(-2)^

ox{-iywe have

Lemma 1.9. If E is a vector bundle on X and if H*'^^{X,E{—j)) = 0 for
every 0 < j < 3, then H*{S, Ejsr) = 0.

Since W is of rank 2, sl{U) is isomorphic to S^U 0 (det U)~^ = (S^W)(1). By
Proposition 1.7 and Lemma 1.9, we have364

S. Mukai

Proposition 1.10. Let S be as above. Then H\S,sl{U)\s) vanishes for every

i,H^{S,{sll(){n)\s) vanishes for every n and U\s or{S®i/)(l)|5 has no nonzero
global sections.

§2. Proof of Theorem 0.2 in the case ^ = 10

Let A be a 3-dimensional subspace of H'^{X,L) and Sa the intersection of
X = Xi6 and the linear subspace P{H^{X,L)/A) o{P{H°{X,L)). Let La andUa be the restrictions of L and U to Sa, respectively. Let E be the subset of theGrassmann variety G{3,H^{X,L)) consisting of A's such that Sa are smooth

K3 surfaces and that the vector bundles Ua are stable with respect to the ample
line bundles La-

Proposition 2.1. E is a nonempty open subset of G{Z, H°{X,L)).
Proof. Ua is a. rank 2 bundle and det Ua = L~^. By Moishezon's theorem[9], Pic Sa is generated by La if A is general. Since H°{Sa,Ua) = 0 by Propo

sition 1.10, Ua is stable if A is general. Since the stableness is an open condition
[8],
we
have
our
proposition.
q.e.d.
In this section we shall prove the following:(2.2) If two 3-dimensional subspaces A and B belong to E and if the polarized
K3 surfaces {Sa,La) and {Sb,Lb) are isomorphic to each other, then Sa andSb, and hence A and B, are equivalent under the action of G.
Let tp : Sa Sb be an isomorphism such that (P*Lb = LaStep I. There is an isomorphism (3 :Ua - ^ ip*UB.

Proof. Since ci{Ua) = -ci{La) and ci{Ub) = -Ci{Lb), the first Chernclasses of Ua and (p*Ub are same. Since {Sb,Ub) is a deformation of {Sa,Ua),
Ub and Ua have the same second Chern number. Hence the two vector bundles'Homos{UA,(p*UB) and Sndos{UA) have the same first Chern class and thesame second Chern number. Therefore, by the Riemann-Roch theorem and ^Proposition 1.10, we havexinomosiUA^T'UB)) = x{£ndos{UA))= x(.OsA + XUHUa))=2.
By the Serre duality, we have

dim Homc)s(f7^,^*C/B) + dim Homc»5(v*17B,CCi)> xC^iomos {UA, <p'UB)) = 2.
Hence there is a nonzero homomorphism from Ua to (P*Ub or vice versa. Since
Ua and <p*Ub are stable vector btmdles and have the same slope, the nonzero
homomorphismisanisomorphism.
q.e.d.

K3 Surfaces and Fano 3-folds

365

Step II. There is an isomorphism 7 : Cq Co (as C-vector spaces) suchthat the following diagram is commutative:

Co (S) Os^

Co <8> Os^ = ^*(Co <S> Osi

Proof. Let 70 be the dual map of■RomiP,Os^) : Romos{v*UB,Os,) —> Homo,(C/A.05j.
Claim: The inclusion UaC Cq iS> Os^ induces an isomorphism Hom(Co, C)
llomos{UA,Os^).
Let K be the dual of the quotient bundle (Cq (8> Ox)/U on X. The natiuralmap from Hom(Co,C) to 'Romoxi^iOx) is an isomorphism because both are

irreducible G-modules. Hence both jH°(X,/C) and H^{XyK) are zero. By theexact sequence0 ^ /C —^ C^ (81 Ox —yO
and Proposition 1.7, we have H*{X,K{—i)) = K;(-t)) = 0 for i = 1,2

and 3. Hence by Lemma 1.9, both and jj^(5,/C|s) are zero and wehave our claim.

By the claim and by applying the claim to <p*Ub C ^*(Co ® C>Sb)> we havea homomorphism 7 : Co —> Co such that the following diagram^omos(CA,Osx]

Homos('P*V^B,Osy

is commutative. Since j3 is an isomorphism, 70 and 7 are isomorphisms and 7
enjoysourrequirement.
q.e.d.

Step III. There is an isomorphism 7 : Co Cq (as C-vector spaces) suchthat (7 ® 1)(I7a) = ^*Ub C Cq % Ox and = 7(2;)^ for every x 6 Co.
Proof. Take an isomorphism 7 which satisfies the requirement of Step II.

Put q{x) = x^ and q'{x) = 7(x)^. Then q and q' are quadratic forms on Co
and both g ® 1 and g' (8> 1 are identically zero on Ua- Hence replacing 7 bysome multiple by a nonzero constant if necessary, we have our assertion by thefollowing:
Claim: The quadratic forms Q on Co such that (Q ® l)|c/^ =0 form at mostone dimensional vector space.

366

S.

Mukai

Let A/" be the kernel of the homomorphism S^a : 5^Co 0 Ox —▶
Since S^Cq is a sum of two irreducible G-modules of dimension 1 and 27 andsince H^{S^a) is a homomorphism of G-modules, we have dim =dim Kerff®(5^Q) = 1. By the exact sequence

H'-'iX.S'Wi-n)) — H'{X,M(-n)) —. H'{X,S%^Ox(-n))
and Proposition 1.7, H*{X,Af{-i)) is zero for every i = 1,2 and 3. Hence by
the Koszul complex (1.8), the restriction map H^{X,//) —> H°{S,/^\s) is
surjective and we have dim H^{S,M\s) < dim H°{X,Af) = 1, which shows ourclaim.

q.e.d.

Step IV. There is an isomorphism 7 : Cq Cq such that (7 0 l)(17yi) =(P*Ub,x^ = 7(2:)^ for every x € Cq and (7 ® l)(xy) = ((7 0 l)(x))((7 0 l)(y))
for every x a and y E.Va-

Proof. Take an isomorphism 7 which satisfies the requirements of Step III.
Then 7 0 1 maps Va onto (p*Vb C Cq 0 Ox and induces an isomorphism F :VaJUa —* ^*{VbIUb) which is compatible with the quadratic forms on VaIUa
and VsJUb' Let ta '■ VaIUa —* sI{Ua) be the restriction of .R : VjU —> sl{U)to Sa- Consider the following diagram:

Va I U A ^ s I { U A )rj.
^ ' i Va l U B ) f ' s l ( V B )The vector bimdles sI{Ua) and sI{Ub) have the quadratic forms / (tr/^)/2
and all the homomorphisms in the above diagram are isomorphisms compatible
with the quadratic forms by Proposition 1.4. If ^ is an automorphism of sI{Ua)and preserves the quadratic form, then g 01 —g comes from an automorphismof Ua because H^{S, Z/2Z) = 0. Since every endomorphism of Ua is a constantmultiplication, g is equal to ±id. Therefore, the above diagram is commutative
up to sign. Hence, for 7 or —7, the above diagram is commutative. Since

xy = rA{y){x) for every x eUa and y E Va, 7 or —y satisfies our requirements,
where

y

E

Va J U a

is

the

image

of

y

e

Va .

q.e.d.

We shall show that, for the isomorphism 7 in Step IV, 7 = 1 ® 7 : Co —> C®
satisfies 7(xy) = 7(x)7(y) for every x,y E Cq. Ux,y E Cq, then xy + yx is equalto 5(x,y), where 6(x, y) is the inner product associated to the quadratic form q.

Hence the realpart of xy is equal to 6(x, y)/2, that is, xy-b{x,y)/2 belongs to Cq.
Since 7 preserves the quadratic form q, 7(x,y) and 7(x)7(y) have the same realpart, that is, their difference belongs to Cq. Put 6{x,y) = 7(x,y) — 7(x)7(y)for
every x, y e Cq. 5 : Cq 0 Cq —* Cq is skew-symmetric and 5 0 1 is identicallyzero on C/a 0 Va C Co 0 Cq 0 .
Step V. 5 0 1 is identically zero on Va 0 Va C Cq (g> Cq 0 •Proof. Since ^® 1 is skew-symmetric and identically zero on Ua<S>Va, 6®1
induces a skew-symmetric form 6 on Va/Ua- Since Va/Ua is isomorphic tosl{UA),/^^{VAlUAy is also isomorphic to sI{Ua) and has no nonzero global

sections. Hence 6 is zero and 5 (gi 1 is identically zero on Va 0 Va. q.e.d.
Step VT. Every homomorphism / from Va to Ua is zero.
Proof Since VaIUa is isomorphic to sI{Ua), there are no nonzero homo-morphisms from VaIUa to Osji• Hence VaIUa cannot be a subsheaf of Cq^Osx ■Therefore, the exact sequence 0 —* Ua —* Va —▶ VaIUa —* 0 does not
split. Hence the restriction f\uJ^ : Ua —* Ua of / to Ua is not an isomorphism.
Since every endomorphism of Ua is a constant multiplication, is zero and/ induces a homomorphism / : Va/Ua —» Ua- Since Va/Ua = sI{Ua), we
have

Eomos{VA/VA,UA) = H''{SA,'KUA)<S>UA)^ H°{Sa,VAB(S'UA]i»LA)Hence by Proposition 1.10, / is zero and / is also zero. q.e.d.
Step VII. 6 is zero.

Proof Let T be the cokernel of the natural injection /\^Va —* 0
OsJ^ - Since 5 0 1 belongs to Homo^ (T, Co 0 Osj^), it suffices to show that
Homo5(r,05^) is zero. There is an exact sequence
2

O—^Va0BA—^T^/\BA—*O,
where Ea is the quotient bundle (Co 0 Osa)/Va and isomorphic to by thebilinear form 6 associated to q. By Step VI, we have Homc?^(V^ <0Ea^ ^Sa) —Homc>5(VA, Ua) = 0. Since A^Ea is an ample line bimdle, Homos(A^Eyi,05^)
is zero. Therefore, by the above exact sequence, Homcj^(T, Os^) is zero, q.e.d.
By Step VII, 1 © 7 is an automorphism of the Cayley algebra C. The auto
morphism of Xis — G/P induced by 1 © 7 maps Sa onto Sb- Hence we have(2.2) and, in particuljir. Theorem 0.2.
§3. Generic K3 surfaces of genus 7,8, and 9
The proof of Theorem 0.2 in the case g = 7,8, and 9 is very similar to
and rather easier thsm the case g = 10 dealt in the previous sections. The(24 - 2p)-dimensional homogeneous spaces X — Xig-2 C (^ = 7^8 and
9) are also generalized Grassmann variety as in the case 5 = 10. In the caseg = S, Xi4 C is the Grassmann variety G(V, 2) of 2-dimensional quotient

spaces of a 6-dimensional vector space V embedded into P(A^ V) by the Plucker

368

S.

Mukai

coordinates. In the case ^ = 9, X C is the Grassmann variety of 3dimensional totally isotropic quotient spaces of a 6-dimensional vector space

V with a nondegenerate skew-symmetric tensor cr 6 where a quotientf : V V' is totally isotropic with respect to a* if ( / (g) /)(cr) is zero in V 0V'.
The embedding Xig c is the linear hull of the composite of the naturalembedding X C G(V,3) and the Pliicker embedding G(F, 3) C P(A^F). In
the case g = 7, X C P^® is a 10-dimensional spinor variety. Let F be a 10dimensional vector space with a non-degenerate second symmetric tensor. The

subset of G(F, 5) consisting of 5-dimensional totally isotropic quotient spacesof F has exactly two connected components, one of which is our spinor variety
X. The pull-back of the tautological line bundle Op{l) by the composite X <-+G(F,5) P(A®F) is twice the positive generator L of PicX and the vector
space H^{X, L) is a half spinor representation of Spin(F), the universal coveringgroups of SO(F). In each case, X is a compact hermitian symmetric space andthe anticanonical class of X is equal to dim X — 2 times the positive generator L
of Pic X (Proposition 1.5 and [2] §16). Moreover, by Proposition 1.5 and an easycomputation, we have that the embedded variety X P(ff®(X, L)) has degree
2g — 2. Hence every smooth complete intersection of X and a linear subspaceof codimension n — 2 (resp. n — 3, n - 1) is the projective (resp. canonical,
anticanonical) model of a K3 surface (resp. curve, Fano 3-fold) of genus g.
Each homogeneous space X = X^g-i has a natural homogeneous vector
bundle £ on it. In the case p = 8, we have the exact sequence(3.1)

0—>:F—^0,

where £ (resp. T) is the universal quotient (resp. sub-) bundle and is of rank 2(resp. 4). In the case g = 7 (resp. 9), we have the exact sequence(3.2) 3 ^ £"^ —*V ®Ox -^ £where £ is the universal maximally totally isotropic quotient bundle with respecttO(T®leV®V<S)Ox and is of rank 5 (resp. 3).
Theorem 0.2 is a consequence of the openness of the stability condition andthe following:

Theorem 3.3. Let S and S' be two K3 surfaces which are complete intersec
tions of X2g-2 C (p = 7,8 and 9) and linear subspaces R and R', respec
tively. Then we have(1) if R is general, then the vector bundle is stable with repsect to Os(l),the restriction of L = Ox(l) to S, and(2) if £\s and £\si are stable with respect to Os(l) and 05'(1) and if S C R
and S' C R' are projectively equivalent, then R and R' are equivalent
under the action ofG on X.
For the proof we need the following property of the vector bundle E — £\s.

K3 Surfaces and Fano 3-folds

369

Proposition 3.4. Let S be a complete intersection of X = ^2^-2 C ^ anda g-dimensional linear subspace and E the restriction of £ to S. Then we have(1) H'{S,sl{E)) = 0 for every i,(2) the homomorphism : V —> H^{S,E) is an isomorphism,(3) in the case g = 7 [resp. 9), the kernel of the homomorphism :52V -> H^{S,S^E) {resp. H°{A^a) : -> H^S,A'^E)) is 1dimensional and generated by a ®\, and(4) in the case g = 7 {resp. 8, resp. 9), E{-1),{A^E){-1), {A^E){-2) or^ {a'^E){—2) {resp. E{—1), resp. E{—1) or (A^£?)(—1)) has no nonzero\

globalsections.

We prove the proposition in the case g = 7. The other cases cire similar.

According to [4], we take Qj = Cj — e,+i, 1 < i < 4, and 05 =64 + 65 as aroot basis of SO(IO). The positive roots are a ± ej,i < j and the conjugacy
class of the maximal parabolic subgroup P corresponds to 05 (or 04). The

homogeneous vector bundles C?x(l)j A'5, s/(5) and S^£ are induced by the ir

reducible representations of the reductive part of P with the highest weights\{e\ -1 f- 65), 61 H 1- 6,-, 61 - 65 and 2ei, respectively. The half p of thesum of positive roots is equal to 4ei + 862 + 263 + 64. Applying Bott's theorem,
we have

Lemma 3.5. {g = 7) The cohomology groups of £{n),{A^£){n),{slS){n) and{S^£){n) vanish except for the following cases:(1) H^{X,£{n)),H\X,{A^S){n)), {X, {S^£){n)) for n > 0 and
H\X,{sl£){n)) for n > 1,
(2) H\X,{A^£){-8)), and(3) H^°{X,S{n)),H'^^{X,{sie){n)) for n < -9 and H^°{X,{A^£){m)),H^^{X,{S^€){m)) for m <-10.
Remark 3.6. In the above case 5 = 7, the 10 roots ei + ej, I < i < j < 5, arecomplementary to P. Their sum is equal to 4(ei + 1- 65) and this is 8 times
the fundamental weight w. By Proposition 1.5, the self intersection number ofC?x(l) is equal to"ill ^ = 10! n {i+i)-' = 12.
peRp

0<t<i<4

Hence X is a 10-dimensional variety of degree 12 in P^® and the anticanonicalclass is 8 times the hyperplane section.

Proof of Proposition 3.4 (in the case 5 = 7): 5 is a complete intersection of
8 members of |C?jf (1)|. Hence, if .A is a vector bundle on X and H*'^°{X, >l(-o))
vanishes for every 0 < a < 8, then so does H*{S, >l|s).

370

S. Mukai(1 and 4) (1) and the vanishings of H'^{S,E{-1)) and H°{S,{A^E){-1))follow immediately from Lemma 3.5. Since is isomorphic to Ox{2), a'^S is

isomorphic to {A^~^Sy ® ^x(2). Hence by the Serre duality and Lemma 3.5,
we have^ H^°-'(X,(A25)(i-8))^ = 0
andH'(X,(A'^5)(-2-i)) = Hi°-'(X,(A''0''(2 + t-8)r^ H^°-'(Jf,f(i-8)))^ = 0,
for every 0 < t < 8. Therefore, {A^E){—2) or (A^jE?)(—2) has no nonzero global
sections.

(2) By the Serre duality, H'(X, JS?^(-i)) and are isomor
phic to 5(i - 8))^ and JT®~'(X,5(i - 8))^, respectively and both are
zero for every 0 < i < 8, by Lemma 3.5. Hence both H°(5, E^) and H^{S, E^)
vanish. Therefore, by the exact sequence (3.2), we have (2).
(3) Let K be the kernel of the homomorphism 5^a : 5^ V (8> Ox —» S^£. We
have the exact sequence

Q—*K-^S^V<^Ox—*S^£—^ 0.
The G-module 5^ V is isomorphic to the direct sum of an irreducible G-moduleof dimension 54 and a trivial G-module generated by <t. Hence the G-module

H°(X,/C) = Ker H^{S^a) is 1-dimensional and generated by cr. By Lemma 3.5
and the Kodaira vanishing theorem, (5^f)(-t)) and if*(X,Gx(-0)
are zero. Hence by the above exact sequence, H^{X,K{—i)) vanishes for every

1 < » < 8. By using the Koszul complex, we have that the restriction map

H^{X,K) -» JT®(5,/C|5) is surjective. Therefore, the kernel of .H'®(5'^a|s) is at
most 1-dimensional. It is clear that the kernel contains <7 01. Hence we have(3).

q.e.d.

Proof of Theorem 3.3: Let S (resp. S') be a K3 surface which is a complete
intersection of X and a linear subspace P (resp. P') and E (resp. E') the

restriction of 5 to 5 (resp. S'). If P is general, then Pic S is generated by0^(1) and, by (4) of Proposition 3.4, E is stable. Hence we have i). Assume

that S and S' are isomorphic to each other as polarized surfaces and that E
and E' are stable. By (1) of Proposition 3.4 and the SEune argument as Step

I in §2, jE; and E' are isomorphic to each other. By (2) of Proposition 3.4, weK3 Surfaces and Fano 3-folds

371

have an isomorphism (3 : V V and a commutative diagram

V<S>Os ^ E —> 0J.}

V 0 Os' ^ J5' —» 0.
Hence, in the case g = 8, S and 5' are equivalent under the action of GL(V). In
the case ^ = 7 or 9, by (3) of Proposition 3.4, 5^/5 maps a to aa for a nonzero
constant a. Hence, replacing ft by we may assume that preserves a.

Hence S and S' are equivalent under the action of SO(V', a) or Sp(y, a), q.e.d.

§4. Generic K3 surface of genus 6
A K3 surface of genus 6 is obtained as a complete intersection in the

Grassmann variety G(2, K®) of 2-dimensional subspaces in a fixed 5-dimensional
vector space F®. G(2,V®) is embedded into P® by Pliicker coordinates andhas degree 5. A smooth complete intersection Xs C P® of G(2, V®) and 3
hyperplanes in P® is a Fano 3-fold of index 2 and degree 5. A smooth complete

intersection and a quadratic hypersurface in P® is an anticanonical divisorof Xi and is a K3 surface of genus 6. The isomorphism clsiss of X5 does notdepend on the choice of 3 hyperplanes and X^ has an action of PGL(2) (seebelow).

Theorem 4.1. Let S and S' be two general smooth complete intersections of

Xs and a quadratic hypersurface in P®. 7/5 c P® and S' C P® are protectively
equivalent, then they are equivalent under the action o/PGL(2) on Xs.
All the Fano 3-folds of index 2 and degree 5 are unique up to isomorphism[5]. There are several ways to describe the Fano 3-folds. The following is mostconvenient for our purpose: Let V be a 2-dimensional vector space and / € 5®Van invariant polynomial of etn octahedral subgroup of PGL(y). / is equal toxy(x^ ~ y^) for a suitable choice of a basis {x,y} of V. Then the closure Xs of
the orbit PGL(V)-f in P^S^V) := (S^V - {0})/C* is a Fano 3-fold of index

2 and degree 5, [11]. ff'^(Xs,Ox(2)) is generated by 7r°(A'5, C?x(l)) = S^V,[5], and has dimension ^(-Kx)^ + 3 = 23. Hence the kernel A of the natural
map 5^H®(X,C>jf(l)) —» H\X,Ox{2)) is a 5-dimensional SL(V)-invariant
subspace. As an SL(V')-module, 5^jH®(A',0;t^(l)) is isomorphic to 5^(5®V') =© S^V © © 1. Hence we have

Proposition 4.2. (1) H^{Xs,Ox{-Kx)) is isomorphic to © 5®V © 1
as S\j{V)-module, and(2) the vector space A of quadratic forms which vanish on Xs C P® is isomorphic to S*V as SL(y)-module.

372

S.

Mukai

There is a non-empty open subset H of | - Kx\ and a natural morphism
S/PGL(V') —> JFg. Both the target and the source are of dimension 19 and themorphism is birational by the theorem. Hence by the proposition we have
Corolleiry 4.3. The generic K3 surface of genus 6 can be embedded into

as an aniicanonical divisor in a unique way up to the action of PGL(V). In
particular, the moduli space Tq is birationally equivalent to the orbit space(S^^V e S^V)/PGL{V).
First we need to show that PGL(V') is the full automorphism group of :Proposition 4.4. The automorphism group AutXs of X^ is connected andthe natural homomorphism PGL(V) —» Aut Xs is an isomorphism.
Proof. There is a 2-dimensional family of lines on X^ C P® and a 1-

dimensional subfamily of lines I of special type, i.e., lines such that Ni/x —0(1) © 0(-l). The union of all lines of special type is a surface and has singu
larities along a rational curve C. C is the image of the 6-th Veronese embeddingof P(V) = P^ into P(5®V). C is invariant under the action of AutX5. Ev
ery automorphism of X5 induces an automorphism of C. Hence we have thehomomorphism a : AutXs —» AutC = PGL(V), Since a|pGL(v) is an isomorphism, AutXs is isomorphic to PGL(V) x Kera. Let g be an automorphism of

JCs which commutes with every element of SL(V). Since 5®V is an irreducible
SL(V)-module, g is the identity by Schur's lemma. q.e.d.
Next we construct an equivariant embedding of A's into the Grassmannvariety G{2,S'^V). Let W be the 2-dimensional subspace of 5^V generated
by -t- and x^y^ for some basis {x,y} of V and Y the closure of the orbit
PGL(V)-[tV] in G{2, S^V). Consider the morphism J : G{2, S'^V) —> P.(5®y)
for whichJ((C, + CM)=det(»^ ^). ^where is the dual basis of {x,y}. Then J is a PGL(V)-equivariant
morphism and sends [W] to the point f,f = xy{x^ — y^). Hence J maps

Y onto Xi C P»(5®V). Define two GL(V)-homomorphisms <p : A^S^V —»S^V 8) (det Vy and j : A^S^V —* 5®V ® det V byip{g A ft) = ^ ijk{DiDjDkg){D..iD^iD^kh) 0{X A F)"®
i,j,k=±la n d,(pAA)=det(^;W373

K3 Surfaces and Fano 3-foIds

where D±i are the derivations by X and Y. The GL{V)-module isdecomposed into the direct sum of irreducible GL(V')-submodules Kery? andKerj. Since = 0, the Pliicker coordinates of W lies in the lineax

subspace P = P*(Ker(/?) of P,(A^5'*V) and Y is contained in the intersection
G{2,S'^V) n P. The morphism J is the composite of the Pliicker embedding

G{2, S^V) C P.(A25^y) and the projection P,(j) : P,{A^S^V) » P^S^V)

from the linear subspace P»{Kery). Since the restriction of P,(j) to P is an
isomorphism, J gives a PGL(y)-equivariant isomorphism from the projective

variety Y C P onto X5 C P,(5®V).
Lemma 4.5. Y coincides with the intersection of G(2,5'^V') and P in
P^A'^S^V).
Proof. Let Y' be the intersection of G{2,5^V) and P and B (resp. B')
the vector space consisting of quadratic forms on P which vanish on Y (resp.
Y'). Both Y and Y' are intersections of quadratic hypersurfaces. Hence itsuffices to show that B = B'. Since G(2,5^V) does not contain P, B' is not

zero. On the other hand, since Y" c P is isomorphic to Xs c P°, P is anirreducible SL(V)-module by Proposition 4.2. As we saw above, Y is contained
PGL(V')-equivariantly in Y' and hence B' is an SL(V')-submodule of B. HenceB'

coincides

with

B.

q.e.d.

So we have constructed a PGL(V')-equivariant embedding of ^"5 intoG(2, S^V) and shown that coincides with the intersection of its linear hulland G{2,S^V).
Proof of Theorem 4.1. There is a universal exact sequence

0S'^V^Ox

—*

P

—*0

on G{2,S*V), where S (resp. P) is the universal sub- (resp. quotient) bundle

0^ and has rank 2 (resp. 3). Let S and S' be two members of the anticanonicalf linecir system | — Kx\ on X^. By the same arguments as in Sections 2 and 3,
we have(i) H*(5,5/(^)|5) = 0 for every t,(ii) If S is general, then the vector bundle is stable with respect to ©5(1),a n d(iii) If 5|s and S\s' are stable with respect to Os(l) and C>s'(l), respectively,
and if S and S' sure isomorphic as polarized surfaces, then there are isomorphisms a : f|5 —^ £\s' and /? G GL(5^V') such that the diagram
0

—^

P|5

ajj

S^V®Os1'/S®!

0 —> PI5. — S'^V^Os-

374

S. Mukai

is commutative. In particular, the automorphism 0 of G{2,S^V) induced
by 0 maps S onto S' isomorphicaliy.

Since is the intersection of C?(2,5^ V) and the linear span of S (resp. 5'),the automorphism 0 maps X5 onto itself. Hence, by Proposition 4.4, S and S'
are equivalent under the action of PGL{V) on X^. q.e.d.
§5. Fano 3-folds of genus 10

In this section we shall prove Theorem 0.9 in the case g = 10. The othercases g = 7^8 and 9 are very similar.

Let V and V be Fano 3-folds which are complete intersections of Xn cand linear subspaces of codimension 2. By the Lefschetz theorem, both
PicF and Pic V are generated by hyperplane sections. Let U be the universal
subbundle of Cq (S> as in Section 1 and F and F' the restrictions of W to V

and V, respectively.

Proposition 5.1. Let ip : V V be an isomorphism. Then <p*{F') is isomorphic to F.

Proof. Let 5 be the generic member of | - Kv\ and put S' = (p{S). The
Picard group of 5 is generated by the hyperplane section. The restrictions

E = F|5 and E' = F'|s' are stable vector bundles as we saw in the proof of
Proposition 2.1. Hence F and F' are also stable vector bundles. Put M =7iomov{E,<p*F'). By Step I in Section 2, there is an isomorphism /o : E
Hence the restriction of M to 5 is isomorphic to EndosiE)- By
Proposition 1.10, we have

H^{S, M(n)|5) = Os{n)) 0 {si E){n)) = 0
for every integer n. Since H^{V,M{n)) is zero, if n is sufficiently negative,
we have by induction on n that H^{V,M{n)) is zero for every n. In par
ticular H^{V,M{-1)) vanishes and hence the restriction map M) —>MI5) is surjective. It follows that there is a nonzero homomorphism/ : E —^ 'P*^' such that f\s = fo- Since fo is an isomorphism, the cokemelof / has a support on a finite set. Since the Hilbert polynomials x{E{n)) andxiifp*E'){n)) are same, the cokernel of f is zero and / is an isomorphism.
q.e.d.

By Proposition 5.1 and similar argmnents as Step II-VII in Section 2, wehave an isomorphism 0 : F (p*F' and an isomorphism 7 : Co —* Cq suchthat the diagramFnip^{F')nCo®Ov ^ Co<S>Ov = 'p'{Co<8>Ov')

K3 Surfaces and Fano 3-folds

375

is commutative and such that 1 © 7 is an automorphism of the Cayley algebra
C. Hence the automorphism of Xis = GjP induced by 1 © 7 maps V onto V\
which shows Theorem 0.9 in the case g = 10.
§6. Curves of genus < 9In this section we shall show the following:
Theorem 6.1. The generic curve of genus < 9 lie on a K3 surface.

In the case 5 < 6, the generic curve is realized as a plane curve C of degreed <6 with only ordinary double points. Take a general plane curve D of degree
6—d and let S be the double covering of the plane with branch locus CUD. Thenthe minimal resolution 5 of 5 is a K3 surface and contains a curve isomorphic
to C.

In the case 6 < p < 9, we shall show that the generic curve C of genus gcan be embedded into P® by the complete linear system of a hne bundle L ofdegree g + 4 and that there is a K3 surface S which is a complete intersectionof 3 quadratic h3rpersurfaces in P® and which contains the image of C.

Let C be a curve of genus 6 < p < 9 and D an effective divisor on C ofdegree p - 6. Put L = u}c^ Oc{-D). Then L is a hne bundle of degree p + 4.

If Z) is general, then dim H^{C,L) = 6. Since degL®^ > degu/c. we have
dim H0(C,£®2) = 2(p + 4) + 1 - p = p + 9.
Proposition 6.2. If C and D are general, then we have

(1) L is very ample and dim H^{C,L) = 6,(2) the natural map
S^H^{C,L) —» jH0(C,L®2)
is surjective and its kernel V is of dimension 12 — g, and(3) there are 3 quadratic hypersurfaces QuQ^ and Qz inP{H°{C,L)) whichcontains the image of C by $|£,| and such that the intersection 5 = Qi n
Q2 n Qz is a K3 surface.

Proof. It suffices to show that there exists a pair of C and D which satisfies

the conditions (1), (2) and (3). Let H be a smooth rational curve of degreep — 4 in P® whose Unear span < R > has dimension p — 4. Since i2 is anintersection of quadratic hypersurfaces, the intersection of 3 general quadratic
hypersurfaces Qi,Q2 and Qz which contain J2 is a smooth K3 surface. Let Cq
be the intersection of S and a general hyperplane H. We show that the pairof the generic member C of the complete linear system |Co + i2| on 5 and thedivisor D = R\c satisfies the conditions (1), (2) and (3).
The intersection number {Co • H) is equal to degH = p - 4 > 2. Hence theUnear system |Co + i2| has no base points. Therefore C is smooth and D iseffective. The genus of C is equal to (Co + RY/2 +1 = p and the degree of D is

376

S.

Mukai

equal to {Co + R.R) = g - 6. Since ojc is isomorphic to Oc{C) = OciCo + R),
the line bundle L = uJc{-D) is isomorphic to Oc{Cq), the restriction of thetautological line bundle of P® to C. There is a natural exact sequence
0 Os{-R) Os{Co) —» Oc{Co) —> 0.

Since H^{S, Os{-R)) = 0 for i = 0 and 1, the restriction map if®(P®,Op(1))
H^{S,Os{Co)) —. H\C,Oc{Co)) is an isomorphism. Hence the morphism $|/,| is nothing but the inclusion map C <—> P® and (1) and (3) cireobvious by our construction of C.

Claim. Let Vq be the vector space of the quadratic forms on P® which are

identiccdly zero on Cq U R. Then the dimension of Vq is at most 12 - g.
Let Fi = 0 be the defining equation of the quadratic hypersurface Qi for
i = 1,2 and 3 and G = 0 that of the hyperplane H. Let F be any quadratic

form on P® which is identically zero on Cq U R. Since F is identicidly zero on
Co, F is equal to aiFi + 03^2 + a^F^ + GG' for some constants ai,a2 and 03
and linear form G'. Since Fi,F2,Fz and F are identically zero on i?, so is GG'.

Hence G' is identically zero on R. Therefore, the vector space Vq is generated
by FiyF2,FQ and GG',G' being all linears from vanishing on < H >. Since

dim < R >— 5 - 4, we have dim Fq < 3 + 5 — (p - 4) = 12 - p.

Since C is a general deformation of Co U R, we have, by the claim, that thedimension of V is also at most 12 - g. Since

dim S^H\C,L)-dim H\C,L^^) = 21 - {g + 9) = 12 - g,
H^{C, is generated by jff°(C, L) and V has exactly dimension 12 - g.
q.e.d.

By the theorem and Corollaries 0.3 and 4.3, we have
Corollary 6.3. The generic curve of genus Z < g < 9 is a complete intersection
in a homogeneous space.

References[1] Borcea, C.: Smooth global complete intersections in certain compact homogeneous complex manifolds, J. Heine Angew. Math., 344 (1983), 65-70.
[2] Borel A. and F. Hirzebruch: Characteristic classes and homogeneous spaces
I, Amer. J. Math., 80 (1958), 458-538: II, Amer. J. Math., 81 (1959), 315382.

[3] Bott, R.: Homogeneous vector bundles, Ann. of Math., 66 (1957), 203-248.
[4] Bourbaki, N.: "Elements de mathematique" Groupes et alg^bre de Lie,
Chapitres 4,5, et 6, Hermann, Paris, 1968.

K3 Surfaces and Fano 3-folds

377[5] Iskovskih, V.A.: Fano 3-folds I, Izv. Adak. Nauk SSSR Ser. Mat., 41 (1977),
516-562.

[6] Iskovskih, Fano 3-folds II, Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978),469-506.

[7] Kodaira, K. and D.C. Spencer: On deformation of complex analytic structures, I-II Ann of Math., 67 (1958), 328-466.
[8] Maruyama, M.; Openness of a family of torsion free sheaves, J. Math.
Kyoto Univ., 16 (1976), 627-637.
[9] Moishezon, B.: Algebraic cohomology classes on algebraic manifolds, Izv.
Akad. Nauk SSSR Ser. Mat., 31 (1967), 225-268.
[10] Mori, S. and S. Mukai: The uniruledness of the moduli space of curves ofgenus 11, ' Algebraic Geometry", Lecture Notes in Math.,n°1016, 334-353,

Springer-Verlag, Berlin, Heidelberg, New York and Tokyo, 1983.
[11] Mukai, S. and H. Umemura: Minimal rational threefolds, "Algebraic Ge
ometry", Lecture Notes in Math., n°1016,490-518, Springer-Verlag, BerUn,

Heidelberg, New York and Tokyo, 1983.
[12] Pijateckii-Shapiro, I and I.R. Shafarevic: A Torelli theorem for algebraic
surfaces of type K3, Izv. Akad. Nauk. SSSR, Ser. Mat., 35 (1971), 503-572.
[13] Saint-Donat, B.: Projective models of K3-surfaces, Amer. J. Math., 96(1974), 602-639.

Shigeru Mukai
Department of Mathematics
Nagoya University
Furo-cho, Chikusa-ku
Nagoya, 464
Japan