CURVES AND K3 SURFACES OF GENUS ELEVEN

Shigeru Mukai
Graduate School of Poly mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-01, Japan
Received May 22,1995

Abstract. For a general curve C of genus 11, its embeddings into K3 surfaces are unique

up to isomorphisms. Such embedding a:C —> S is constructed as a non-abelian analogueof the duality between the Picard and Albanese varieties. Let S be the moduli space ofstable rank two vector bundles E of canonical determinant on C with hP{E) > 7 . Then S
is a smooth K3 surface and the determinant line bundle hdet is a polarization of genus 11.

The K3 surface S which contains C is the unique 2-dimensional component of the moduli
space of stable rank two sheaves of determinant hdet on S.

In [11], we have begun our study of the Brill-Noether locus

Mc(2, Kj n) = {E\h^{E) > n + 2}/isomin the moduli space Mc(2, K) of stable rank two vector bundles E of canonical determinant
over a curve, i.e., a compact Riemann surface C. In the workshop at Sanda, we discussed

Mc(2, K, 3) for a curve C of genus seven, for which see the forthcoming article. Here, instead,
we study another interesting case:

Theorem 1. For a general curve C of genus eleven, the Brill-Noether locus Mc(2,K,5) isa smooth K3 surface and the restriction hdet of the determinant line bundle of Mc{2, K) isa polarization of genus eleven, i.e., (h^) = 20.

Let {X, h) be a pair of a K3 surface and a line bundle h of degree 20 and Mx{2, h, n) be

the moduli space of stable sheaves £ on X with r(£) = 2, detf ~ ft and xiB) = 2 +n
' . By6], Mx (2, ft, n) is smooth and of dimension 2(11 - 2n) if it is not empty. Hence Mx (2, ft, n)
is a surface only if n = 5 and, in fact, X ;= Mx(2,h,b) is a K3 surface if it is coiripact.
A

A

Moreover, there is a line bundle h of degree 20 on X and (X, h) is isomorphic to (X, h) ([7],Theorem 1.4).

Let S be Mc(2, X, 5) in the theorem and U the Poincare bundle onCxS. We normalize

U so that deiU ~ Kc s hdet (see §4) and restrict it to fibres in the other direction. Namely
we consider the family of vector bundles Up := U\p^s on S parametrized by C.

Theorem 2. Let C, S = Mc{2, K,b), hdet and U he as above. Then the vector bundle Up
on S is stable with respect to hdet and belongs to Ms[2,hdet,^) for every peC. Moreover,the classification morphism a : C —> Ms(2, hdet, h),p [Up], is an embedding.

Let Mil and be the moduli spaces of curves and polarized K3 surfaces (X, h) of genus

11, respectively. Let Vn be the P^Lbundle \h\ over and <Pn : Vn " ■ ^ Mn

the rational map which associates the isomorphism class for every O E \h\ (cf [5]) By

the theorem, the rational map: Afu-- - ^ Vn, [C] ^ iMs{2,hd.„b),a(C)), satisfies
0,1 o 0 = id. Since Vn is irreducible and of the same dimension as Mn, 4>n is birational
and ip is its inverse. Therefore, we have

Unite mixte de I'institute Henri Poincare, CNRS Universite Pierre et Marrie Currie, Paris.
189

190

S. MukaiCorollary 1. Ms(2, hdeu 5), with S = Mc(2, K, 5), is the unique K3 surface which contains
C.

Since Mw is unirational by [4], we also have
Corollary 2. Tu is unirational.

Thus Mc(2,/C,5) and the morphism a : C —> Ms(2,hdeu^) are similar to the Picard

variety and Albanese map X —> AlbX = (Pic® respectively. The morphism a is theK3 hull in the following sense:

Definition An embedding a : C —> A of a curve C into a variety A is a, K3 hull if1) there exist a line bundle L on A and its global sections Si, • • • , s„_i such that L\c — Kc
and C is the complete intersection 5i = • • • = Sn_i = 0 in A,2) every embedding of C into a K3 surface is isomorphic to the restriction of a to thecommon zero locus of a codimension one subspace of < Si, • • • , s„_i >C i/®(A, L), and3) there is an exact sequence20 ^ H\Ncia ® Kc') -4 H'(Ka') /\ H'(Oc),
where the first map is the coboundary map of the long exact sequence associated to thenatural exact sequence

0 —> Tc —> Ta\C —^ ^c/A —^ 0] ^I

and the second is the dual of the Wahl map /\^ H^(Kc) —> H^(Kq) (see [13]).
The K3 hull exists for every general curve of genus p > 7. It is the symmetric space describedin [9] for g = 7,S and 9, and C itself for ^ = 10 and ^ > 12 (cf. [3], [8], [5]). This will be
discussed elsewhere.

Let C; be the locus of curves with in the moduli space Mn and C a general memberof Gg. C is embedded into by \Kc^~^\ and the quadric hull S of its image C P® is aK3 surface of degree 8, where ^ is a. gl of C. There exists a family {5x}xe5 of vector bundles
0 —^ Os[C - A) —> ^ 4(A) ^ 0

of Schwarzenberger type on S parametrized by 5, where A is a hyperplane section of 5 C P^.
Theorem 3. For a general member C of the Brill-Noether locus Mci2, K,b) is smoothand isomorphic to S by the correspondence S 3 x ^ Ex\c ^ A/c(2, K, 5).

The quadric hull S in the theorem is the unique K3 surface which contains the hexagonal

curve C. In fact, if a K3 surface contains G, then the gl on G extends to a line bundle on it.

Another interesting divisor of Adn is the locus G\. For a general member G of Gg,

S = Mc(2,Ar,5) is still a K3 surface and contains a line D, i.e., degfideilz? = 1, whichparametrizes the extensions 0 —> C —^ FJ —> KcC ^ ^ 9 with hP(E) = 7, where ^ isa gl of G. The moduli space Ms(2, hdeu 5) is the unique quartic surface which contains theimage of ^KcC-^ ' ^ ^We prove the theorem in §3 after some preparations in §§1 and 2, and Theorem 1 and 2

in the final section.

Notations P,K and PV are the two projective spaces associated to a vector space V.

The former parametrizes one-dimensional subspaces and the latter quotient spaces. 5" V isthe n-th symmetric tensor product of a vector space or vector bundle V. For an Gx-module

Curves and K3 surfaces of genus eleven

191

S and an Oy-module the tensor product of their pull-backs on the product X x K isdenoted hy

1. Preliminary

Next two lemmas are useful to analyze and construct vector bundles in Mc{2,K,n).
Lemma 1. Let E he a rank two vector bundle of canonical determinant and ^ a. line bundle

on C. If C is generated by global sections, then we have

dimHom<p^(C,£7) > hP{E) — degCThe proof is an easy exercise of the base-point-free pencil trick (see [10] Proposition 3.1).
Lemma 2. Let ^ be a line bundle and consider non-trivial extensions
0——>E—>7]—>0
of ^ by its Serre adjoint rj.

1) The extensions E with hP{E) = hPi^) + h^in) o-re parametrized by the projective spaceP^Coker[52ifO(77) ^2) i4ssu?7ie that the multiplication map S^H^(r}) —> is surjective. Then h^{E) <h^{^)-\- h^{rj) — 1 for every non-trivial extension E. Moreover, the non-trivial extensions Ewith h^{E) = -h h^(r}) - 1 are parametrized by the quadric hull of the image of :C —> F*H°{tj). More precisely, for every point x of the quadric hull, there is the uniqueextension E such that the image of the linear map H^(E) —> is the codimension one
subspace corresponding to x.

Proof. Let e e Ext^ be the extension class and Se : H^(r)) —> coboundary

map. The condition that h^(E) = h^(^)-\- hP{r}) is equivalent to Jg = 0, that is, e lies in the
kernel of the linear map

By the Serre duality, the linear map A is the dual of the multiplication map <S>

H^(r)) —)> Hence Ker A is the dual of Co\ier[S'^H^{r]) —y which shows (1).
The first assertion of (2) is a direct consequence of (1). The condition that h^(E) = hP(^) +h^(ri) — 1 is equivalent to rank^g = 1 by our assumption. There exists a nonzero linear map
a: ^ C such that Jg is the composite of a and its dual : C —> H^{r}Y ^ H\^).
Let x be the point o{F*H^{r)) corresponding to a. Then h = KeT[S'^H^{r]) —> thedegree 2 part of the homogeneous ideal of %{C), vanishes at x, since S^a is the composite ofS'^H^(r}) —^ and the linear map H\rf) —y C corresponding to e by construction.

Thus E with h^{E) = h^(C)-\-h^{r}) — 1 determines a point in the quadric hull. Conversely, apoint in the quadric hull determines e with rank = L Such e i.s unique since A is injective
by our assumption. □See [11] §4 for the following criterion of smoothness:

Proposition 1. Let E be a member of Mc(2, K) with h^(E) = n-y-2 and put a = SofC) 3 — (n-b 2)(n-}-3)/2. Then we have

1) dun[E]Mc{2, K,n) > a, and

2) Mc(2, K, n) is smooth and of dimension o at E if and only if the. m.ultiplication map

S H^(E) —y H^[S'^E) is injective.

S. Mukai

2 . H e x a g o n a l c u r v e o f g e n u s 11

Let 5 C P® be a smooth complete intersection of three quadrics. 5 is a K3 surface by
the adjunction formula and the Lefschetz theorem. Throughout this and next sections weassume that S contains a normal elliptic curve B of degree 6 but no bisecant lines of B.
Such a surface exists by the following:

Lemma 3. Let B C be a normal elliptic curve of degree 6. Then the intersection of
three general quadrics passing through B is a smooth surface and does not contain a bisecant

line of B CP^.
Proof. The first assertion follows from Bertini's theorem, since H C is an intersectionof quadrics. All 5's which contain B are parametrized by a non-empty open subset of theGrassmannian G(3, H^{Ib(2))), which is of dimension 3(9-3) = 18. Let ^ be a bisecant line.
All S"s which contain both B and i are parametrized by an open subset of the Grassmannianwhich is of dimension 15. Since the bisecant lines are parametrized by asurface, we have the second assertion. □Let C C 5 be a smooth member of the complete linear system |A + B|, where A is a

hyperplane section of 5 C P®. Such C exists since \A-\- B\ is free from base points. Since(A2) = 8, {A.B) = 6, = 0 and (A -h Bf = 20,
C is of genus 11. We denote the restriction of Os[B) and Os(A) to C by ^ and r;, respectively.
By the adjunction formula, the canonical line bundle Kc of C is the tensor product of ^ and

Tj. By the exact sequences

0 ^ Os{-A) Os{B) ^ ^ 0
and0 —> Os{-B) Os{A)and by the lemma below, the restriction maps H^{Os(B)) —> B[^(^s(^)) —>H^(rj) are isomorphisms. In particular, |^| is a g^ and its Serre adjoint r? is a ^^4. The
embedding G 5 ^ P® is given by the complete linear system \r].
Lemma 4. W{Os{A - B)) = W{Os(B - A)) = 0 for every i.

Proof. Since {B - A.A) < 0, we have H^{Os{B - A)) = and hence H'^(Os(A -B)) =0

by the Serre duality. Since (A - Bf = -4, we have x(Os(B - A)) = 0. Hence it suffices toshow H^(Os(A — B)) = 0. Assume the contrary and let D be a member of \A — B\. Since(D^) = -4 and (D.A) = 2, D is a union of two disjoint lines. Since {D.B) = 6, one of themmeets B at at least three points. This is a contradiction. □Tensoring Os{A) with the exact sequence

0 ^ Os{-B) —> Of —> Os{B) 0,
we have

Lemma 5. The multiplication map

WiOsiB)) ® iT'iOsiA)) —> H\Os(A + B))
is an isomorphism.

Curves and K3 surfaces of genus eleven

193

The multiplication map S'^H^(Os{A)) —>• H''(0s{2A)) is surjective and its kernel is ofdimension 3. Since S is smooth, every quadric passing through S is smooth at x G 5.
Therefore, we have

Lemma 6. The multiplication map S^H^(Ix(A)) —H^{Os(2A)) is injective for every
X e S, where Ix is the maximal ideal ofOs at x.
The non-existence of bisecant lines in S implies the following:
Lemma 7. For an effective divisor D, we have1) h°(^(D)) = 2 if D is of degree < 3, and2) h°{^{D)) + h°(r](—D)) < max {8 - d,d+ 2}, where d is the degree of D.

Proof If degL> < 2, then h^{7]{-D)) = h°(rj) - degT), since l?;! is very ample. Since^7? = Kc, we have hP{^{D)) = = 2 by the Riemann-Roch theorem. Assume that

degD = 3 and /i°(f(D)) > 2. Then D is contained in a trisecant line ^ of C C P^. This is acontradiction since ^ is contained in S and (LB) = (tC - A) = 2. So we have proved (1).
By the Riemann-Roch theorem, we haveh'^i^iD)) -hh'{v(-D)) = 2h'iaD)) ^4-d.
Hence (2) immediately follows from (1) if d = 2 or 3. If d > 4, then h^(^(D)) < d - 1 by(1). Hence we have (2). □Now we are ready to prove the following:

Proposition 2. Let E be a semi-stable rank two vector bundle of canonical determinant on
C. Then we have1) h^{E) < 1, and2) if hP{E) = 7, then E is stable and contains a line subbundle isomorphic to f or ^{p)for a point p G C.

Proof. If hP(E) < 7, then there is nothing to prove. Hence we may assume that h^(E) > 7.

By Lemma 1, E contains a subsheaf isomorphic to ^ and we have an exact sequence0 ^ ^(D) -^E-^ ni-D) 0

for an effective divisor D of degree < 4. Hence we have h%E) < h^{^{D)) + h^(Tj(-D))) < 1
and degT> = 0 or 1 by Lemma 7, which shows (1) and the second assertion of (2). Assume

that E is not stable and let 0 —)■ a —)■ E —> P —>> 0 be an exact sequence withdego deg/? 10. Then either o or is isomorphic to ^(7)) for an effective divisor Dof degree 4. Hence we have h\E) < h^i^D)) + h'{7](-D)) < 6 by Lemma 7. This is acontradiction and E is stable. □3. Proof of Theorem 3

Let B and C C 5 C P® be as in the previous section. (5, Os{A -h B)) is a polarized
K3 surface of genus 11. We study Ms{2, A H- B, 5) and their restrictions to C, using vector
bundles of Schwarzenberger type. Let

H'{S,SxVo,{Ix(A),Os(B))) => Ex4+/(7,(.4),Os(B)),

194

S. Mukaibe the local-global spectral sequence of extension groups, where Ix is the ideal of a point xof 5. By Lemma 4, the natural map(1)

Ext\,^mA),Os{B))~C

is an isomorphism. Let

(2)

0

Os{B)

UA)

0.

be the unique non-trivial extension of Ix{A) by Os{B). Since H^{Os{B)) = 0, we have theexact sequence

0 ^ H\Os[B)) H\Ex) —> H\lx{A)) 0.
The following is obvious:

Lemma 8. 1) dim H^{Sx) = 7 and Ex is generated by global sections.
2) W{Ex) = 0 fori = 1,2.
Since the three linear mapsi)

S''H^(Os{B))—>H^(0s(2B)),^ii) H'(Os(B)) ® H\Ix(A)) H%Ix{A + B)) andhi) S''H^(IxiA)) H\ll{2A)).
are injective by Lemma 5 and 6, we have

Lemma 9. The natural linear map S'^H^{Ex) —> H^(S'^Ex) is injective.
Let Ex be the restriction of the vector bundle Ex to the curve C. Since det^x ~ Os{C),
we have the exact sequence

O-^E^-^Ex^Ex^O.
By Lemma 8 and the Serre duality, we have

Lemma 10. The restriction map —> H^(Ex) is an isomorphism.

In particular, dim (Ex) = 7 and Ex is generated by global sections. Restricting theabove (2) to the curve C, we have the exact sequence

(3)

0

——>

Ex

—>r}

—>0

and

^

(4) 0 H\() H\Ex) H'(Ix{A)) —f 0
if X ^ C. If X 6 C, then /x(i4) (g) Oc contains the sky-scraper sheaf k{x) as a subsheaf andthe torsion-free quotient is isomorphic to rji-x). Hence we have the exact sequence

(5)

0

^(x)

—^Ex^

vi-^)

0-

Proposition 3. Ex is stable for every x e S.

Proof. Assume that x^C. Since h^E^) < + h''{r]) by (4), the exact sequence (3)does not split. Let C be a line subbundle of E^. If the composite C^ n is zero,then degC = degf < 10. Otherwise C is isomorphic to ri(-D) for a nonzero effective divisor
D If deg D = 1 and D = (p) for a point p € C, then the extension class e of (3) lies in

the kernel of ExtJ,^(p,?) —+ ExtJ,^(p(-p),C(p)). Hence the point corresponding to e in theway of Lemma 2 is p. This is a contradiction. Hence we have degZ) > 2. Since h (E) - 7,
we have deg L> > 5 by Lemma 7. Therefore Ex is stable.

Claim: The exact sequence (5) does not split. , , .

Assume the contrary and let s : Ex —> ^(2:) be a splitting. Let T be the kernel 01

the composite Ex ^ Ex i(x). H^(E) is a 5-dimensional subspace of H^(Ex) andI

Curves and K3 surfaces of genus eleven195

mapped onto H^{Ix(A)). Since ci{!F) = 0, the image of the evaluation homomorphism
0 Os —> ^ C is of rank one. Hence the image is isomorphic to Ix{A), which is acontradiction.

Assume that x £ C and let C he a line subbundle of Ex- We may assume that ^ ~rj(-x - D) for an effective divisor D. D is nonzero by the claim and Ex is stable by Lemma
7.

□

By the proposition, Sx is also stable (with respect to Os{C)) and every endomorphism isa constant multiplication. Hence, by the exact sequence

0 —> si Ex—^ S% S^Ex —^ 0,

the restriction map H^(S^Ex) —> H^(S^Ex) is injective. By Lemma 9 and 10, we have
Lemma 11. The natural linear map S^H^(Ex) —> H^{S^Ex) is injective.

Now we prove Theorem 3. Let T be a copy of S and A be the diagonal of 5 x T. By theLeray spectral sequence, we have an isomorphism

ext'o^iUia'AlOsM^'B)))

Ot(B

-

A)

whose fibres are (1), where a and r are the projections of 5 x T onto S and T, respectively.
Hence we have the universal extension

(6)

0

^

U{a*A

-I-

t*{B

-

A))

0,

whose restriction to 5 x a; is (2) for every x e T. The restriction of .F to C x T is a

Poincare bundle of the family {Exjx^T- By Lemma 8, 10 and Proposition 3, we obtain theclassification morphism Be : S —> Mc(2, K, 5). Be is injective by the following:
Lemma 12. dimHom(^, = 1 for every x e S.

Proof. Assume the contrary. Then Ex contains two distinct subsheaves isomorphic to

Let L be the subsheaf generated by them. If L is of rank one, then L is isomorphic to ^(D)for an effective divisor D and h^{L) > 3, which contradicts Lemma 7. If L is of rank two,then L is isomorphic to ^ 0 f, which contradicts Lemma 11. □Let E he a, member of Mc(2, K, 5). By Proposition 2, there is an exact sequence
0 ——> E —>7] —>0or

0 —> ^(p) —> E —> T]{-p) —> 0

for a point p£C. In the latter case, E is isomorphic to Ep by (1) of Lemma 2 and the claimin the proof of Proposition 3. In the former case, there exists a point x e S such that theimage of H^iE) is H^mA)) by (2) of Lemma 2. By the stability of E, x does

not belong to C. Hence E is isomorphic to Ex. Therefore, Be is surjective. Mc(2,K,5) issmooth and of dimension 2 by Proposition 1 and Lemma 11.

4. Proof of Theorem 1 and 2

Let C be a general curve of genus 11. C does not have a pJq. Hence we have h^{E) < 6 forevery strictly semi-stable rank two vector bundle E of canonical determinant on C. SinceMc(2, K, 5)'s form a proper family when C varies, they are smooth K3 surfaces by Theorem
3.

196

S. Mukai

Proposition 4. There exists a vector bundle U on C X Mc(2, K, 5) such that l{\cx[E] — E
for every E G Mc(2,K,b) and detU ~ Kc 0 hdet- Such U, called the normalized Poincare
bundle, is unique up to isomorphism.
Proof. The moduli space Mc(2, K) is the quotient of an open subset of a Quot scheme byan action of PGL{v). Let B! be the inverse image of Mc{2, K, 5) by the quotient morphism
R —^ Mc(2,K) and U the restriction of the universal quotient bundle. By Proposition 2,h^[E) = 7 for every E G Mc(2,K,b). Hence the direct image ttr'M is a vector bundle ofrank 7. The direct image 7rRf^{U ^c) is of rank 40. Following [12], we consider thevector bundle

ii ®T^%{{deiT^R>jAY^®dei{'KRi^{U^Oc Ec))~^)The action of a central element t G C GL(u) on the three factors are t, andHence this tensor product has an action of PGL{i/) and descends to a Poincare bundle U on
CxMc(2,K,^).

Since Mc(2, K, 5) is regular, there exists a line bundle L on Mc(2, K, 5) such that detU ~Kc S L. Since <^Oc E^c — bi ®Om ^ ^[R'-emMY ^ 0 Kc) (ttmM) 0 L-'
by the (relative) Serre duality. Hence we have(7) hdet = (det R^ttmM) 0 (det ® N~^,
where we put N = detT^MM- Therefore, the universal bundle U ®Om ® K~^) satisfies
our requirement (cf. [1] p. 582). The normalized Poincare bundle is unique since universal
bundles differ only by Pic M(2, K, 5) and the Picard group has no 2-torsion. □The determinant line bundle hdet is ample by [2]. We compute its degree. In the hexagonalcase, T\cxt is a universal family. Since det{!F\cxT) — Kc^O{B -A) by the exact sequence(6), we have hdet - Ot{7(B - A)) 0) (detn(/'|cxT))~^ by (7). By Lemma 10 and (6), we
have the exact sequence

0 ^ H^(Os(B)) 0Ot-^ r^iJ'lcxT) -4 H^(Os(A)) 0 Ot{B - A)-A Ot(B) 0
and det(r»J"|cxr) - Ot(^B - 6A). Hence hdet is isomorphic to Ot[^A - 3B), which is aline bundle of degree 20. So we have proved Theorem 1.

For a hexagonal curve C,U = (T\cxt)^0t['^A — 2B) is the normalized Poincare bundle.

The restriction Up is stable for every p G C by Proposition 3 and a is an embedding by
Theorem 3. Since the properties to be shown are stable by small deformations, we have
Theorem 2.

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[13] Wahl, J.: Introduction to Gaussian maps on algebraic curves, in Complex Projective Geometry, London
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