arXiv:math/0112283v2 [math.AG] 26 Jan 2002 A SUPERSINGULAR K3 SURFACE IN CHARACTERISTIC 2 AND THE LEECH LATTICE I. DOLGACHEV AND S. KOND ¯ O ABSTRACT. We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This configuration is isomorphic to the config- uration of points and lines on the projective plane over the finite field of 4 elements. The surface admits a finite automorphism group isomorphic to PGL(3, F 4 ) · 2 such that a subgroup PGL(3, F 4 ) acts on the configura- tion of each set of 21 smooth rational curves, and the additional element of order 2 interchanges the two sets. The Picard lattice of the surface is a reflective sublattice of an even unimodular lattice II 1,25 of signatuire (1, 25) and the classes of the 42 curves correspond to some Leech roots in II 1,25 1. I NTRODUCTION Let k be an algebraically closed field of characteristic 2. Consider F 4 ⊂ k and P 2 (F 4 ) ⊂ P 2 (k). Let P be the set of points and let ˇ P be the set of lines in P 2 (F 4 ). Each set contains 21 elements, each point is contained in exactly 5 lines and each line contains exactly 5 points. It is known that the group of automorphisms of the configuration (P , ˇ P ) is isomorphic to M 21 · D 12 where M 21 ( ∼ = PSL(3, F 4 )) is a simple subgroup of the Mathieu group M 24 and D 12 is a dihedral group of order 12. In this paper we prove the following theorem: 1.1. Main Theorem. There exists a unique (up to isomorphism) K 3 sur- face over k satisfying the following equivalent properties : (i) The Picard lattice of X is isomorphic to U ⊥ D 20 ; (ii) X has a jacobian quasi-elliptic fibration with one fibre of type ˜ D 20 ; (iii) X has a quasi-elliptic fibration with the Weierstrass equation y 2 = x 3 + t 11 ; Research of the first author was partially supported by NSF grant DMS 9970460 and the Japanese Society for Promotion of Science. Research of the second author is partially supported by Grant-in-Aid for Exploratory Research 11874004. 1
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A SUPERSINGULAR K3 SURFACE IN CHARACTERISTIC 2AND THE LEECH LATTICE
I. DOLGACHEV AND S. KONDO
ABSTRACT. We construct a K3 surface over an algebraically closedfield of characteristic 2 which contains two sets of 21 disjoint smoothrational curves such that each curve from one set intersectsexactly 5curves from the other set. This configuration is isomorphic to the config-uration of points and lines on the projective plane over the finite field of4 elements. The surface admits a finite automorphism group isomorphicto PGL(3, F4) ·2 such that a subgroup PGL(3, F4) acts on the configura-tion of each set of 21 smooth rational curves, and the additional elementof order 2 interchanges the two sets. The Picard lattice of the surface isa reflective sublattice of an even unimodular latticeII1,25 of signatuire(1, 25) and the classes of the 42 curves correspond to some Leech rootsin II1,25
1. INTRODUCTION
Letk be an algebraically closed field of characteristic 2. ConsiderF4 ⊂ kandP2(F4) ⊂ P2(k). LetP be the set of points and letP be the set of linesin P2(F4). Each set contains 21 elements, each point is contained in exactly5 lines and each line contains exactly 5 points. It is known that the groupof automorphisms of the configuration(P, P) is isomorphic toM21 · D12
whereM21(∼= PSL(3, F4)) is a simple subgroup of the Mathieu groupM24
andD12 is a dihedral group of order 12.In this paper we prove the following theorem:
1.1. Main Theorem. There exists a unique (up to isomorphism)K3 sur-face overk satisfying the following equivalent properties:
(i) The Picard lattice ofX is isomorphic toU ⊥ D20 ;(ii) X has a jacobian quasi-elliptic fibration with one fibre of typeD20 ;(iii) X has a quasi-elliptic fibration with the Weierstrass equation
y2 = x3 + t11;
Research of the first author was partially supported by NSF grant DMS 9970460 andthe Japanese Society for Promotion of Science.
Research of the second author is partially supported by Grant-in-Aid for ExploratoryResearch 11874004.
(iv) X has a quasi-elliptic fibration with 5 fibres of typeD4 and the groupof sections isomorphic to(Z/2)4 ;
(v) X contains a setA of 21 disjoint (−2)-curves and another setB of21 disjoint (−2)-curves such that each curve from one set intersectsexactly 5 curves from the other set with multiplicity1;
(vi) X is birationally isomorphic to the inseparable double coverof P2
with branch divisor
x0x1x2(x30 + x3
1 + x32) = 0;
(vii) X is isomorphic to a minimal nonsingular model of the quartic sur-face with7 rational double points of typeA3 which is defined by theequation
x40 + x4
1 + x42 + x4
3 + x20x
21 + x2
0x22 + x2
1x22 + x0x1x2(x0 + x1 + x2) = 0;
(viii) X is isomorphic to the surface inP2 × P2 given by the equations
x0y20 + x1y
21 + x2y
22 = 0, x2
0y0 + x21y1 + x2
2y2 = 0.
The automorphism group Aut(X) contains a normal infinite subgroup gen-erated by168 involutions and the quotient is a finite group isomorphic toPGL(3, F4) · 2.
Since the group PSL(3, F4) is not a subgroup of the Mathieu groupM23,our theorem shows that the classification of finite groups acting symplec-tically on a complex K3 surface due to S. Mukai [Mu] does not extend topositive characteristic.
2. SUPERSINGULAR K3 SURFACES IN CHARACTERISTIC2
2.1. Known facts. Recall that asupersingularK3 surface(in the sense ofShioda) is a K3 surface with the Picard group of rank 22. This occurs onlyif the characteristic of the ground fieldk is a positive primep. By a resultof M. Artin [Ar], the Picard latticeSX = Pic(X) of a supersingular K3surface is ap-elementary lattice (i.e. the discriminant groupS∗
X/SX is ap-elementary abelian group). The dimensionr of the discriminant group overFp is even and the numberσ = r/2 is called theArtin invariant. We shallassume thatp = 2. A fundamental theorem of Rudakov and Shafarevich[RS1],[RS2] tells that any supersingular K3 surface admitsa quasi-ellipticfibration, i.e. a morphismf : X → P1 whose general fibre is a regularbut not smooth geometrically irreducible curve of genus 1. Over an opensubsetU of the base, each fibre is isomorphic to an irreducible cuspidalcubic, the reducible fibres are Kodaira genus 1 curves of additive type. Theclosure of the set of cusps of irreducible fibres is a smooth irreducible curveC, thecusp curve. The restriction off to C is a purely inseparable coverof degree 2. It follows thatC is isomorphic toP1, hence by adjunction is
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 3
a (−2)-curve onX. A surface with a quasi-elliptic fibration is unirational.Thus any supersingular K3 surface in characteristic 2 is unirational. This isone of the main results of [RS1],[RS2].
In this paper we study a supersingular K3 surface with the Artin invari-ant σ = 1. It follows from the classification of 2-elementary lattices ofsignature(1, 21) that the Picard lattice of such a K3 surface is unique up toisometries (see [RS2],§1). Therefore it is isomorphic toU ⊥ D20. Here,as usual,U denotes the unique even unimodular indefinite lattice of rank 2andAn, Dn, En denote the negative definite even lattices defined by the op-posite of the Cartan matrix of the simple root system of the correspondingtype. Moreover it is known that any supersingular K3 surfacewith Artininvariantσ = 1 is unique up to isomorphisms ([RS2],§11).
2.2. The Weierstrass equation.
2.3. Proposition. LetX be a supersingularK3 surface whose Picard lat-ticeSX is isometric toU ⊥ D20. ThenX has a quasi-elliptic fibration withone singular fibre of typeD20 and a section. Its Weierstrass equation is
y2 + x3 + t11 = 0.
Proof. First we shall see that the above Weierstrass equation defines a K3surface whose Picard lattice is isomorphic toU ⊥ D20. The reducible fibreslie over points at which the affine surface has a singular point. Taking thederivatives we find that such point is given byx = t = 0. This givest = 0as the only solution for ”finite”t. Whent = ∞, we make the substitutionsy → t6y, x → t4x and thent = 1/τ . Then we have the equation
y2 + x3 + τ = 0
which is nonsingular overτ = 0. The discriminant∆ of the fibration hasorder 20 att = 0. Hence the fibration defines a K3 surface (see [RS2],§12, [CD], Chapter 5). It has a unique reducible fibre. It follows from theclassification of fibres of quasi-elliptic fibrations that itmust be of typeD20.Obviously its Picard lattice containsU ⊥ D20 which is generated by theclass of components of fibres and a section. Sinceσ ≥ 1, the Picard latticeisomorphic toU ⊥ D20. Now the assertion follows from the uniqueness ofsupersingular K3 surface with the Artin invariant 1.
3. LEECH ROOTS
In this section, we denote byX the K3 surface whose Picard lattice isisomorphic toU ⊥ D20.
4 I. DOLGACHEV AND S. KONDO
3.1. The Leech lattice. We follow the notation and the main ideas from[Bo], [Co1], [Co2], [Ko]. First we embed the Picard latticeSX
∼= U ⊥ D20
in the latticeL = Λ ⊥ U ∼= II1,25, whereΛ is theLeech latticeandU is thehyperbolic plane. We denote each vectorx ∈ L by (λ, m, n) whereλ ∈ Λ,andx = λ + mf + ng, with f, g being the standard generators ofU , i.e.f 2 = g2 = 0, and〈f, g〉 = 1. Note thatr = (λ, 1,−1− 〈λ,λ〉
2) satisfiesr2 =
−2. Such vectors will be calledLeech roots. We denote by∆(L) the set ofall Leech roots. Recall thatΛ can be defined as a certain lattice inR24 =RP
1(F23) equipped with inner product〈x, y〉 = −x·y8
. For any subsetA ofΩ = P1(F23) let νA denote the vector
∑
i∈A ei, wheree∞, e0, . . . , e22 isthe standard basis inR24. A Steiner systemS(5, 8, 24) is a set consistingof eight-element subsets ofΩ such that any five-element subset belongs toa unique element ofS(5, 8, 24). An eight-element subset inS(5, 8, 24) iscalled anoctad. ThenΛ is defined as a lattice generated by the vectorsνΩ − 4ν∞ and 2νK , whereK belongs to the Steiner systemS(5, 8, 24).Let W (L) be the subgroup generated by reflections in the orthogonal groupO(L) of L. Let P (L) be a connected component of
P (L) = x ∈ P(L ⊗ R) : 〈x, x〉 > 0.
ThenW (L) acts naturally onP (L). A fundamental domain of this actionof W (L) is given by
D = x ∈ P (L) : 〈x, r〉 > 0, r ∈ ∆(L).
It is known thatO(L) is a split extension ofW (L) by Aut(D) ([Co2]).
3.2. Lemma. LetX be theK3 surface whose Picard latticeSX is isomor-phic to U ⊥ D20. Then there is a primitive embedding ofSX in L suchthat the orthogonal complementS⊥
X is generated by some Leech roots andisomorphic to the root latticeD4.
Proof. Consider the following vectors inΛ:
X = 4ν∞ + νΩ, Y = 4ν0 + νΩ, Z = 0,(3.1)
T = (x∞, x0, x1, xk2, ..., xk22
) = (3, 3, 3,−1,−1,−1,−1,−1, 1, ..., 1)(3.2)
whereK = ∞, 0, 1, k2, ..., k6 is an octad. The corresponding Leech roots
x = (X, 1, 2), y = (Y, 1, 2), z = (0, 1,−1), t = (T, 1, 2)(3.3)
generate a root latticeR isomorphic toD4. ObviouslyR is primitive inL.Let S be the orthogonal complement ofR in L. Then it is an even lattice
of signature(1, 21) and the discriminant group isomorphic to(Z/2Z)2 (thediscriminant groups ofS andR are isomorphic). Since such a lattice isunique, up to isometry, we obtain thatS ∼= U ⊥ D20.
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 5
3.3. (42 + 168) Leech roots. We fix an embedding ofSX into L as inLemma 3.2. LetP (X) be the positive cone ofX, i.e. a connected compo-nent of
x ∈ P(SX ⊗ R) : 〈x, x〉 > 0
which contains the class of an ample divisor. LetD(X) be the intersectionof D andP (X). It is known that Aut(D(X)) ∼= M21 · D12 whereM21
∼=PSL(3, F4) and D12 is a dihedral group of order 12 ([Bo],§8, Example5). Each hyperplane boundingD(X) is the one perpendicular to a negativenorm vector inSX which is the projection of a Leech root. There are twopossibilities of such Leech rootsr:
(i) r andR (∼= D4) generate a root lattice isomorphic toD4 ⊥ A1;(ii) r andR generate a root lattice isomorphic toD5.
3.4. Lemma. There are exactly forty-two Leech roots which are orthogo-nal toR.
Proof. Let R′ be the sublattice ofL generated by vectorsx, y, z from (3.3).ObviouslyR′ is isomorphic to the root latticeA3. By Conway [Co1], theLeech roots orthogonal toR′ are
4ν∞ + 4ν0, νΩ − 4νk, 2νK ′,
whereK ′ contains∞, 0 andk ∈ 1, 2, ..., 22.Among these Leech roots, the followings are orthogonal to vector t ∈ R:
4ν∞ + 4ν0, νΩ − 4νk, 2νK ′,
whereK ′ ∈ S(5, 8, 24), K ′ ∩ K = ∞, 0 or K ′ ∩ K = ∞, 0, 1, ∗ andk ∈ K \ ∞, 0, 1. As is easy to see the number of these roots is equal to42.
3.5. Remark. In his paper [To], Todd listed the759 octads of the SteinersystemS(5, 8, 24) (p.219, Table I). The octads in the proof of Lemma 3.4correspond to the followingEα, Lβ in his table. Here we assume
3.6. Lemma. There are exactly168 Leech rootsr such thatr andR gen-erate a root latticeD5.
Proof. We count the number of such Leech rootsr with 〈r, t〉 = 1. Suchrcorresponds to one of the following vectors inΛ (see the proof of Lemma3.4):
νΩ − 4νk (k /∈ K), νK ′ (K ′ ∈ S(5, 8, 24), | K ∩ K ′ |= 4, 1 /∈ K ′).
Obviously the number of vectors of the first type is 16. Since the numberof octads containing fixed 4 points is 5, the number of vectorsof the secondtype is 40. Thus we have the desired number56 × 3 = 168.
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 7
3.7. The 42 smooth rational curves. Let
w = (0, 0, 1)
be theWeyl vectorin L (characterized by the property that〈w, λ〉 = 1 foreach Leech root). Since the Leech lattice does not contain(−2)-vectors,〈w, r〉 6= 0 for any (−2)-vectorr in L = Λ ⊥ U . Consider its orthogonalprojectionw′ to R⊥ ⊗ Q. Easy computation gives
w′ = w + 5z + 3x + 3y + 3t ∈ R⊥,
and
〈w′, w′〉 = 14.(3.4)
Fix an isometry fromSX to R⊥ and leth be the divisor class correspondingto w′. The above property ofw implies that〈h, r〉 6= 0 for any (−2)-vectorsr in SX . Composing the embedding with reflections in(−2)-vectorsin SX , we may assume thath is an ample divisor class. Each of the 42Leech vectors from Lemma 3.4 defines a vectorv from Pic(X) with self-intersection−2. Since〈h, v〉 = 1, by Riemann-Roch, we obtain thatv isthe divisor class of a curveRv with R2
v = −2. Sinceh is ample, eachRv isan irreducible curve, and hence isomorphic toP1.
Thus the 42 Leech roots in Lemma 3.4 define 42 smooth rational curvesin X.
3.8. Lemma. Let X be a K3 surface over a field of characteristic 2. Thefollowing properties are equaivalent:
(i) The Picard latticeSX of X is isomorphic toU ⊥ D20;(ii) X has a quasi-elliptic fibration with five singular fibres of typeD4 and
sixteen disjoint sections;(iii) There are two familiesA andB each consisting of21 disjoint smooth
rational curves. Each member in one family meets exactly fivemem-bers in another family. The setA ∪ B generatesSX .
Proof. (i) ⇒ (ii) Fix an isometry fromSX to R⊥. Consider the five disjoint(−2)-curvesKk corresponding to
νΩ − 4νk, (k ∈ 2, 3, 5, 14, 17).
Then | 2K2 + R1 + R2 + R3 + R4 | gives a genus 1 fibration with fivesingular fibers of typeD4, whereRi correspond to the vectorsEi in Remark3.5. Moreover the curves corresponding to the vectorsLi are 16 disjointsections of this fibration. The fibration satisfies the Rudakov-Shafarevichcriterion of quasi-elliptic fibration [RS2],§4.
(ii) ⇒ (i) It follows from the Shioda-Tate formula [Sh1] that the discrim-inant ofSX is equal to 2.
8 I. DOLGACHEV AND S. KONDO
(ii) ⇒ (iii) We take forA the set of sections and multiple components offibres. We take forB the set of non-multiple components of reducible fibresand the cusp curve. The curves from the setA correspond to the LeechrootsEα, 4ν∞+4ν0, and the curves fromB correspond to the Leech rootsLα, νΩ − 4νk. The cusp curveC correspond to the Leech root4ν∞ + 4ν0
(see the following Remark 3.10). It follows from the Shioda-Tate formulathat the setA ∪ B generatesSX .
(iii) ⇒ (ii) Take R0 ∈ A and 4 curvesR1, R2, R3, R4 from B whichintersectR0. Then
F = 2R0 + R1 + R2 + R3 + R4(3.5)
defines a quasi-elliptic fibration with a fibreF of type D4. Let N be thefifth curve fromB which intersectsR0. It is easy to see that the curves fromA ∪ B \ N which do not intersect the curvesR1, R2, R3, R4 form fourmore reducible fibres of typeD4. The remaining curves give 16 disjointsections.
3.9. Lemma. LetX be theK3 surface whose Picard lattice is isomorphicto U ⊥ D20 andA,B are the sets described in the previous lemma. Leth be the divisor class corresponding to the projection of the Weyl vectorw = (0, 0, 1) ∈ Λ ⊥ U to R⊥ ⊗ Q. Then
h = ΣR∈A∪BR/3.(3.6)
Proof. Observe that the right-hand sideh′ of (3.6) intersects each curveRi
with multiplicity 1. Since the curvesRi generateSX , h − h′ is orthogonalto SX and hence is equal to zero.
3.10. Remark. Note that the curveC from the proof of Lemma 3.8 is thecusp curveC ′ of the quasi-elliptic fibration. In fact, since it intersects thedouble components of the reducible fibres, it is a 2-section.Since the com-ponentsFi of fibres andC ′ generate a sublattice of finite index inSX , wemust haveC ≡ C ′ +
∑
i niFi for some rational coefficientsni. Intersectingthe both sides with eachFj we get
∑
i ni(Fi · Fj) = 0. Since the inter-section matrix of irreducible components is semi-definite,this implies that∑
i niFi = mF , whereF is the class of a fibre andm ∈ Q. But now(C − C ′)2 = −4 − 2(C · C ′) = (mF )2 = 0 givesC ∼ C ′. Since a smoothrational curve does not move, we getC = C ′.
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 9
3.11. Remark. A quasi-elliptic fibration with 5 fibres of typeD4 has 16sections if and only if the Picard lattice has discriminant equal to 4, andhence is isomorphic toU ⊥ D20. This follows immediately from theShioda-Tate formula [Sh1]. Also, ifX has a quasi-elliptic fibration withfive fibres of typeD4 and sixteen sections, then the sixteen sections are au-tomatically disjoint (note thatm-torsion sections on an elliptic or a quasi-elliptic surface over a field of characteristic dividingm are not necessarilydisjoint). Hence the surface contains 42 smooth rational curves with in-tersection matrix as described in Lemma 3.8. Let us show this. Let C bethe cuspidal curve of the fibration and letFi = 2E0(i) + E1(i) + . . . +E4(i), i = 1, . . . , 5, be the reducible fibres of the fibration. The divisorD = 2C + E0(1) + . . . + E0(4) is nef and satisfiesD2 = 0. Thus it de-fines a genus 1 pencil|D|. No sections intersectD, so they are contained infibres of the fibration. Also the componentsE1(5), E2(5), E3(5), E4(5) donot intersectD and hence are contained in fibres of|D|. We have 16 sec-tions, so there existsEi(5), sayE1(5), such that it intersects at least four ofthe sections. By inspection of the list of possible reducible fibres of a genus1 fibration we find thatE1(5) together with 4 disjoint sections form a fibreof typeD4. Now we have 12 remaining sections which intersect one of thecomponentsEi(5), i = 2, 3, 4. Again we may assume thatE2(5) intersectsat least four of the remaining 12 sections, and hence we find another fibreof typeD4. Continuing in this way we obtain that the 16 sections togetherwith the curvesE1(5), E2(5), E3(5), E4(5) form four fibres of typeD4. Inparticular, they are disjoint.
3.12. The 168 divisors. Let r be a Leech root as in Lemma 3.6. The pro-jectionr′ of r into SX ⊗Q is a(−1)-vector. We can directly see that eachr′
meets exactly 6 members in each familyA andB stated in Lemma 3.8. Forexample, if we use the notation as in Remark 3.5 and take(νΩ − 4ν4, 1, 1)asr, thenr meets exactly twelve Leech roots corresponding to
We remark that these twelve curves are mutually disjoint. Weset
l = (2h +∑
R∈A
R)/7.(3.7)
Thenl ∈ SX , l2 = 2, l ·R = 0 for anyR ∈ A, l ·R = 1 for anyR ∈ B and
2r′ = 2l − (R1 + · · · + R6),
whereR1, ..., R6 are(−2)-curves inA which meetr′. Eachr′ defines anisometry
sr′ : x → x + 2(r′ · x)r′
10 I. DOLGACHEV AND S. KONDO
of SX which is nothing but the reflection with respect to the hyperplaneperpendicular tor′.
Consider a quasi-elliptic fibration onX with five reducible fibres of typeD4 with 16 sections. LetC be the cusp curve and letF be the class ofa fibre. We now take asA the set of 21 curves consisting ofC and simplecomponents of fibres, and asB the set consisting of 16 sections and multiplecomponents of fibres. Then we can easily see that
l = C + F, sr′(l) = 5l − 2(R1 + · · · + R6),(3.8)
sr′(Ri) = 2l − (R1 + · · · + R6) + Ri.
4. EXPLICIT CONSTRUCTIONS
4.1. Inseparable double covers. Let L be a line bundle on a nonsingularsurfaceY over a field of characteristic2, ands ∈ H0(Y,L2). The pair(L, s) defines a double coverπ : Z → Y which is given by local equationsz2 = f(x, y), where(x, y) is a system of local parameters andf(x, y) = 0is the local equation of the divisor of zeroes of the sections. Replacingswith s + t2, wheret ∈ H0(Y,L), we get an isomorphic double cover. Thesingular locus ofZ is equal to the zero locus of the sectionds ∈ H0(Y, Ω1
Y ⊗L2). It is locally given by the common zeroes of the partials off(x, y). Thecanonical sheaf ofX is isomorphic toπ∗(ωY ⊗ L). All of these facts arewell-known (see, for example, [CD], Chap. 0).
We shall consider the special case whenY = P2 andL = OP2(3). Thesections is identified with a homogeneous formF (x0, x1, x2) of degree 6.We assume thatF6 = 0 is a reduced plane curve of degree 6. The expectednumberN of zeroes of the sectiondF6 is equal to the second Chern numberc2(Ω
1P2(6)). The standard computation givesc2 = 21. So, if N = 21, the
partial derivatives of any local equation ofF6 at a zerox of ds generatethe maximal idealmx. This easily implies that the coverZ has 21 ordinarydouble points. So a minimal resolutionX of Z has 21 disjoint smoothrational curves.
4.2. The double plane model. To construct our surfaceX as in the MainTheorem 1.1, (vi), we take the inseparable cover corresponding to the planesexticF6 = 0 with dF6 vanishing at the 21 points ofP2 defined overF4.Consider the sextic defined by the equation
F6 = x0x1x2(x30 + x3
1 + x32) = 0.(4.1)
Direct computations show that the partials ofF6 vanish at the setP of 21points ofP2(F4):
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 11
(1, 1, a2), (1, a, 1), (1, a2, 1), (1, a, a), (1, a, a2), (1, a2, 1), (1, a2, a2),
whereF4 = 0, 1, a, a2. Thus the exceptional divisor of a minimal resolu-tion X of the corresponding double coverX ′ → P2 is a setA of 21 disjointsmooth rational curves. LetΣ(P) be the blow up of the setP. ThenXis isomorphic to the normalization of the base changeX ′ ×P2 Σ(P). Nowlet P be the set of 21 lines onP2 defined overF4. Their equations area0x0 + a1x1 + a2x2 = 0, where(a0, a1, a2) ∈ P. The 21 lines are dividedinto three types: three lines which are components of the sextic F6 = 0; 9lines which intersect the sextic at three of its double points; 9 lines whichintersect the sextic at two points of its double points. In the last case eachline is a cuspidal tangent line of the cubic :x3
0 + x31 + x3
2 = 0. The properinverse transforml of any line inΣ(P) is a smooth rational curve with self-intersection−4 which is either disjoint from the proper inverse transform ofthe sextic, or is tangent to it, or is contained in it. In each case the pre-imageof l in X is a(−2)-curve taken with multiplicity 2. LetB be the set of suchcurves. Since each linep ∈ P contains exactly 5 points fromP, and eachpointp ∈ P is contained in exactly 5 lines fromP , the setsA andB satisfyproperty (v) from the Main Theorem. Finally notice that the mapX → P2
is given by the linear system|l|, wherel is defined by (3.7).
4.3. A switch. A switchis an automorphism ofX which interchanges thesetsA andB. Let us show that it exists. Consider the pencil of cubic curvesgenerated by the cubicsx0x1x2 = 0 andx3
After blowing up the base points, we obtain a rational elliptic surfaceVwith 4 reducible fibres of typeA2. The 12 singular points of the four fibresare the pre-images of the points inP not appearing in (4.2). The surfaceXis the inseparable cover of the blow-up ofV at these points. It has an ellipticfibration with 4 fibres of typeA5. The base of this fibration is an inseparabledouble cover of the base of the elliptic fibration onV . Now we find thatthe Mordell-Weil group is isomorphic toZ/2Z ⊕ (Z/3Z)2. Nine of thesesections are curves from the setA of (−2)-curves corresponding to pointsin P2(F4). They correspond to the base points (4.2). Another 9 sectionsare from the setB of (−2)-curves corresponding to lines inP2(F4). Theycorrespond to the lines dual to the points (4.2), i.e. the linesax0 + bx1 +cx2 = 0, where(a, b, c) is one of (4.2). Fix a zero sections0 represented bysay aA-curve. LetFi =
∑
k∈Z/6 Ek(i), i = 1, 2, 3, 4, be the reducible fibres
12 I. DOLGACHEV AND S. KONDO
such thatEk(i)·Ek+1(i) = 1 ands0 intersectsE0(i). It is easy to see that the9 sections fromA intersect the componentsE0(i), E2(i), E4(i), three eachcomponent. TheB-sections intersect the componentsE1(i), E1(i), E5(i),again three each. Also, the componentsE0(i), E2(i), E4(i) areB-curvesand the componentsE1(i), E1(i), E5(i) areA-curves. Thus we obtain thatA ∪ B consists of 18 sections and 24 components of fibres. Now considerthe automorphismT of the surfaceX defined by the translation by the 2-torsion sections1. Obviously,s1 intersects the componentsE3(i). We seethatT interchanges the setsA andB. Also, note that ifs0 corresponds to(a0, a1, a2), thens1 is aB-curve corresponding to the linea0x0 + a1x1 +a2x2 = 0. To be more specific, lets0 correspond to the pointp = (1, 1, 0).Thens1 corresponds to the linel : x0 + x1 = 0. Indeedl is the cuspidaltangent ofF : x3
0 + x31 + x3
2 = 0 at the pointp. After we blow-upp, thetwo sectionss0 ands1 intersect at one point of the fibre represented byF .But this could happen only for the 2-section since the 3-torsion sections donot intersect (we are in characteristic 2). Similarly we seethatT sends asection represented by a point(a, b, c) to the section represented by the dualline ax0 + bx1 + cx2 = 0. Also, it is easy to see thatT transforms a fibrecomponent corresponding to a point to the component corresponding to thedual line. ThusT is a switch.
We remark that we can directly find a configuration of four fibres of typeA5 and 18 sections on 42 smooth rational curvesA ∪ B. This implies theexistence of such elliptic fibration (and hence a switch) without using theabove double plane construction.
4.4. Mukai’s model. In a private communication to the second author, S.Mukai suggested that the surfaceX is isomorphic to a surface inP2 × P2
defined by the equations:
x20y0 + x2
1y1 + x22y2 = 0, x0y
20 + x1y
21 + x2y
22 = 0.(4.3)
To prove this we consider the linear system|h|, whereh is the divisor classrepresenting the projectionw′ of the Weyl vectorw ∈ L. Recall thath isan ample divisor satisfyingh · R = 1 for any R ∈ A ∪ B. Consider thequasi-elliptic fibration|F | as in Lemma 3.8. We may assume that the setA consists of non-multiple components of fibres and the cusp curveC, andthe setB consists of 16 sections, and 5 multiple components of fibres.
Let D1 = 2R0 + R1 + R2 + R3 + R4 be a reducible member of|F | andlet S1, . . . , S4 be four sections intersectingR1. Consider the quasi-ellipticpencil |F ′| = |2R1 + S1 + S2 + S3 + S4|. The set(B \ R0) ∪ C isthe set of irreducible components of reducible members of|F ′|. The curveC ′ = R0 is its cuspidal curve. Now we check that, for anyR ∈ A ∪ B,
(h − C − F ) · R = (C ′ + F ′) · R.
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 13
Since the curvesR generate the Picard group ofX, we see that
h = (C + F ) + (C ′ + F ′).(4.4)
The linear system|C + F | defines a degree 2 mapπ1 : X → P2 whichblows down the curves from the setA and maps the curves from the setBto lines. The linear system|C ′ + F ′| defines a degree 2 mapπ2 : X → P2
which blows down the curves from the setB and maps the curves fromAto lines. Letφ : X → P8 be the map defined by the linear system|h|.Using (4.4) one easily sees thatφ mapsX isomorphically onto a surfacecontained in the Segre varietys(P2 × P2). Let us identifyX with a surfacein P2 × P2. The restriction of the projectionsp1, p2 : P2 × P2 → P2 are themapsπ1, π2 defined by the linear systems|C + F | and|C ′ + F ′|. Let l1, l2be the standard generators of Pic(P2 × P2) and[X] = al21 + bl1 · l2 + cl22 bethe class ofX in the Chow ring ofP2 × P2. Intersecting[X] with l21 andl22,we geta = c = 2. Since,h2 = 14 we get
This shows thatX is a complete intersection of two hypersurfacesV1, V2 ofbidegree(2, 1) and(1, 2).
The image of each curve from the setA (resp. B) in P2 × P2 is a linecontained in a fibre of the projectionp1 (resp. p2). The fibres of the pro-jectionp1 : V1 → P2 define a linear system|L| of conics inP2. Let V bethe Veronese surface parametrizing double lines. The intersection|L| ∩ Vis either subset of a conic, or is the whole|L|. Since|L| contains at least 21fibres which are double lines which are not on a conic, we see that |L| ⊂ V .Thus all fibres ofp1 : V1 → P2 are double lines. This implies that theequation ofV1 can be chosen in the form
A0y20 + A1y
21 + A2y
22 = 0,
where the coefficients are linear forms inx0, x1, x2. It is easy to see thatthe linear forms must be linearly independent (otherwiseX is singular).Thus, after a linear change of the variablesx0, x1, x2, we may assume thatAi = xi, i = 0, 1, 2.
Now consider a switchT ∈ Aut(X) constructed in section 4.3. Obvi-ously it interchanges the linear systems|C + F | and|C ′ + F ′| and hence isinduced by the automorphismss of P2 ×P2 which switches the two factors.This shows thats(V1) = V2, hence the equation ofV2 is
y0x20 + y1x
21 + y2x
22 = 0.
14 I. DOLGACHEV AND S. KONDO
We remark that the curves
(x0, x1, x2) = (a0, a1, a2), a0y20 +a1y
21 +a2y
22 = 0, (a0, a1, a2) ∈ P2(F4)
and their images under the switch form 42 smooth rational curves onX =V1 ∩ V2 satisfying property (v) in the Main theorem.
4.5. The quartic model. Consider the quartic curve inP2 defined by theequation
F4(x0, x1, x2) =(4.5)
x40 + x4
1 + x42 + x2
0x21 + x2
0x22 + x2
1x22 + x0x1x2(x0 + x1 + x2) = 0.
This is a unique quartic curve defined overF2 which is invariant with respectto the projective linear group PGL(3, F2) ∼= PSL(2, F7) (see [Di]). LetYbe the quartic surface inP3 defined by the equation
x43 + F4(x0, x1, x2) = 0(4.6)
Clearly the group PGL(3, F2) acts onY by projective transformations leav-ing the planex3 = 0 invariant.
By taking the derivatives, we find thatY has 7 singular points
We denote byPi1≤i≤7 be the set of these singular points. Each singularpoint is locally isomorphic to the singular pointz4 + xy = 0, i.e. a rationaldouble point of typeA3. LetX be a minimal resolution ofY . We claim thatX is isomorphic to our surface. Note that the points inP2 whose coordinatesare the first three coordinates of singular points (4.7) are the seven points ofP2(F2). Leta0x0+a1x1+a2x2 = 0 be one of the seven lines ofP2(F2). Theplanea0x0+a1x1+a2x2 = 0 in P3 intersectsY doubly along a conic whichpasses through 3 singular points ofY . For example, the planex0+x1+x2 =0 intersectsY along the conic given by the equations
x20 + x2
1 + x22 + x2
3 + x0x1 + x0x2 + x1x2 = x0 + x1 + x2 = 0.
We denote byC ′i1≤j≤7 the set of these conics and byI(j) the set of indices
i with Pi ∈ C ′j.
Let R(i)1 +R
(i)2 +R
(i)3 be the exceptional divisor of a singular pointPi. We
assume thatR(i)2 is the central component. It is easy to check that the proper
inverse transformCj of each conicC ′j in X intersectsR(j)
2 with multiplicity1, if Pj ∈ C ′
i, and does not intersect other components. This easily gives
h = 2Cj +∑
i∈I(j)
(R(i)1 + 2R
(i)2 + R
(i)3 ),(4.8)
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 15
whereh is the pre-image inX of the divisor class of a hyperplane sectionof Y , and we identify the divisor classes of(−2)-curves with the curves.
Observe now thatX contains a setA of 21 disjoint smooth rationalcurves: seven curvesCi and 14 curvesR(i)
1 , R(i)3 . Each of the seven curves
R(i)2 intersects exactly 5 curves from the setA (with multiplicity 1). We
shall exhibit the additional 14 smooth rational curves which together withthese seven curves form a setB of 21 disjoint(−2) curves such thatA ∪ Bsatisfies property (v) of the Main Theorem.
To do this let us take a line inP2(F2), for example,x1 = 0. Then thereare exactly four points onP2(F2) not lying on this line:
and intersectsY along a quartic curveQ′ given by the equations
x40 + x4
2 + x20x
21 + x2
1x22 + x2
2x20 + x0x1x2(x0 + x1 + x2) = x1 + x3 = 0.
It splits into the union of two conics
Q′1 : x2
0 + ax22 + x0x1 + ax1x2 = x1 + x3 = 0,
Q′2 : x2
0 + a2x22 + x0x1 + a2x1x2 = x1 + x3 = 0.
Let C ′ be one of the seven quartic curvesC ′i’s corresponding to the line
x1 = 0. It is given by the equations
x20 + x2
2 + x23 + x0x2 = x1 = 0.
Then H meetsC ′ at q1 = (1, 0, a, 0), q2 = (1, 0, a2, 0). Note thatQ′1
passes throughP1, . . . , P4, q1 andQ2 passes throughP1, . . . , P4, q2. Eachsingular pointPi of the quartic is locally isomorphic toz4 + xy = 0 andthe local equation of the quarticQ′ at this point isz = 0. This easily showsthat the proper inverse transform ofQ′ in X consists of two smooth rationalcurvesQ1 andQ2 each intersects simply the exceptional divisor at one pointlying in different components ofR(i)
1 ∪ R(i)3 . ThusQ1 (or Q2) intersects
exactly 5 curves fromA: 4 curvesR(i)j (j = 1 or 3, i = 1, . . . , 4) and the
proper inverse transform ofC ′ in X. Also, it is clear that 14 new quarticcurves obtained in this way neither intersect each other after we resolve thesingularities of the quartic nor intersect the curvesR
(i)2 . This proves the
claim.
16 I. DOLGACHEV AND S. KONDO
4.6. Remark. The configurationC of 14 curvesCi, R(i)2 is isomorphic to
the configuration of points and lines inP2(F2). The group PGL(3, F2) actson the surfaceX via its linear action inP3 leaving the hyperplanex3 = 0fixed. Its action on the configurationC is isomorphic to its natural action onlines and points.
5. AUTOMORPHISMS OFX
In this section we describe the group of automorphisms of thesurfaceX. First we exhibit some automorphisms ofX and then prove that theygenerate the group Aut(X).
5.1. The group PGL(3, F4). Consider the double plane model ofX withthe branch curveF6 = 0 as in (4.1). The groupG = PGL(3, F4) actsnaturally in the planeP2(F4). For anyg ∈ G, let Pg = F6(g(x)). Wecan see thatdF (g(x)) = (dF )(g(x))g′(x) vanishes at the same set of 21points onP2(F4). One checks that the linear system of curves of degree 5vanishing at the 21 points is spanned by the curves
x40x1 + x4
1x0 = 0, x40x2 + x4
2x0 = 0, x42x1 + x4
1x2 = 0.
By integrating we see thatPg = αgF6 + A2g, whereAg is a cubic form and
αg ∈ k∗. It is easy to see that the mapχ : g → αg is a homomorphismof groups. The restriction ofχ to the subgroupG′ = PSL(3, F4) is trivial,since the latter is a simple group. Also, it is directly verified that the branchcurve does not change under the transformations given by thediagonal ma-trices [a, 1, 1] which together withG′ generateG. Hence, for allg ∈ G,αg = 1. Thus the mapTg : (z, x) → (z + Ag(x), g(x)) is an automorphismof the double plane. It is easy to verify thatAg′g(x) = Ag′(g(x)) + Ag(x).SinceTg′g(z, x) = (z + Ag′g(x), g′g(x)) and
we obtain that the mapg → Tg is a homomorphism. SinceG is simple, it isinjective.
Note that the induced action ofG on the setsA andB is isomorphic tothe action ofG on points and lines inP2(F4).
5.2. The 168 Cremona transformations. Let P1, . . . , P6 be 6 points inP2(F4) such that no three among them are colinear. Since each smoothconic overF4 contains exactly 5 points (it is isomorphic toP1
F4), we see
that the setP = P1, . . . , P6 is not on a conic. This allows one to definea unique involutive quintic Cremona transformationΦ given by the linearsystem of curves of degree 5 defined overF4 with double points at the pointsof P (see [Coo], Book IV, Chapter VII,§4). The transformationΦ blowsdown each conic to a point ofP2(F4). Let B : F6 = 0 be the branch
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 17
curve (4.1) of the double coverX → P2. By adding toF6 the square of acubic form, we may assume that each point fromP is a double point of thecurveB. SinceΦ−1(a line) is a quintic with double points atP, the imageB′ = Φ(B) of B is a curve of degree5 · 6− 4 · 6 = 6. The image of a conicthrough 5 points fromP is a double point ofB′. Let F ′
6(y0, y1, y2) = 0be the equation ofB′ and letΦ be given by homogeneous polynomials ofdegree 5:
whereqi(x0, x1, x2) = 0 are the equations of the exceptional conics. Takingthe partials we find
2∑
i=0
∂F ′6
∂yi
∂fi
∂xj
=∂F6
∂xj
q + F6∂q
∂xj
, j = 0, 1, 2,
whereq =∏6
i=1 qi. Let P ∈ P2(F4) \ P. We know that the partials ofF6
vanish atP . Adding toF6 a square of a cubic polynomial we may assumethatF6(P ) = 0. Since the determinant of the jacobian matrix( ∂fi
∂xj) of Φ is
invertible outside of the locusq = 0, we obtain that the partials ofF ′6 vanish
at Φ(P ). This shows that the partials ofF ′6 vanish at all points ofP2(F4).
Using the same argument as in 5.1 we find thatF ′6 = αF6 +F 2
3 . SinceΦ2 isthe identity, we getα2 = 1 and henceα = 1. Now we can define a birationaltransformation of the double plane by the formulaΦ(z, x) = (z+F3, Φ(x)).This birational automorphism extends to a regular automorphism (sinceXis a minimal model).
Next let us show that the number of setsP is equal to 168. We say that asubset ofP2(F4) is independentif no three points from it are colinear. LetNk be the number of independent subsets ofP2(F4) of cardinalityk. Wehave
Finally we remark that the above 168 automorphisms act on thePicardlatticeSX as reflections with respect to 168(−4)-vectors stated in subsec-tion 3.12. This follows from equations (3.8).
We denote byN the normal subgroup of Aut(X) generated by 168 invo-lutions.
18 I. DOLGACHEV AND S. KONDO
5.3. The automorphism group. It is known that the natural map fromAut(X) toO(SX) is injective ([RS2],§8, Proposition 3). Moreover, Aut(X)preserves the ample cone, and hence Aut(X) is a subgroup of the fac-tor groupO(SX)/W (SX)(2), whereW (SX)(2) is the group generated by(−2)-reflections. By the argument in [Ko], Lemma 7.3, we can see that,for any isometryg in O(SX) preserving an ample class, there exists anautomorphismϕ ∈ N such thatg ϕ ∈ Aut(D(X)). This implies thatO(SX)/W (SX)(2) is a subgroup of a spilt extension ofN by Aut(D(X)).Recall that Aut(D(X)) ∼= M21 · D12 (see subsection 3.3). HereM21 =PSL(3, F4). The Frobenius automorphism ofF4 gives an involution on 21lines and 21 points inP2(F4). This involution induces an isometryι of SX
because 42 smooth rational curves generateSX (Lemma 3.9). The dihe-dral groupD12 is generated byι, a switch and an automorphism of order 3induced from a projective transformation ofP2(F4) given by
a 0 00 1 00 0 1
.
We shall show thatι can not be represented by an automorphism ofX. Todo this we consider a quasi-elliptic fibration with five fibresof typeD4 fromLemma 3.8. Thenι fixes three fibres of typeD4 and switches the remainingtwo fibres. This does not happen ifι is realized as an automorphism.
Thus we conclude that
Aut(X) ∼= N · PGL(3, F4) · 2,
where the involution 2 is generated by a switch.
5.4. Corollary. The finite group PGL(3, F4) ·2 is maximal in the followingsense. LetG be a finite group of automorphisms ofX, thenG is conjugateto a subgroup of PGL(3, F4) · 2.
Proof. Note thatG fixes the vector
h =∑
g∈G
g∗(h)
which is non-zero becauseh is an ample class andG is an automorphismgroup. The vectorh is conjugate to a vector inD(X) underN (see sub-section 5.3). This means thatG is conjugate to a subgroup of Aut(D(X)).Now the assertion follows.
5.5. Conjecture. Let X be a K3 surface over an algebraically closed fieldk of characteristicp admitting the group PGL(3, F4) · 2 as its group of au-tomorphisms. Thenp = 2 andX is isomorphic to the surfaceX from theMain Theorem.
A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 19
6. PROOF OF THEMAIN THEOREM
6.1. By definition,X satisfies property (i) (see subsection 2.1). By Propo-sition 2.3,X satisfies properties (ii) and (iii). By Lemma 3.8,X satisfiesproperties (iv) and (v). Property (vi) from section 4.2. Property (vii) wasproven in 4.5 and property (viii) in 4.4. The group of automorphisms wascomputed in section 5. The uniqueness follows from the fact that the Artininvariantσ is equal to 1.
It remains to prove the equivalence of properties (i)-(viii).(ii) ⇒ (i). The components of fibres and the zero section define the sub-
lattice U ⊥ D20 ⊂ Pic(X) of rank 22. If the assertion is false, Pic(X)would be unimodular. However, there are no even unimodular lattices ofrank 22 of signature(1, 21).
(i) ⇒ (ii) An isotropic vectorf from U can be transformed with thehelp of(−2)-reflections into the class of a fibre of a genus 1 fibration. Lete1, . . . , e20 be a positive root basis ofD20. . Without loss of generality wemay assume thate1 is effective. This will imply that allei’s are effective.Sincef · ei = 0 for all i’s, the irreducible components ofei’s are containedin fibres of the genus 1 fibration. Since the rank of the subgroup of Pic(X)generated by irreducible components of fibres is at most 20, we see that allei’s are irreducible(−2)-curves. They are all contained in one fibre whichmust be of typeD20. If f ′ is an isotropic vector fromU with 〈f, f ′〉 = 1,the class off ′ − f gives the class of a section. A theorem from§4 of [RS2]implies that the fibration is quasi-elliptic.
(i) ⇔ (iv) Follows from Remark 3.11.(iv) ⇔ (v) Follows from Lemma 3.8.(ii) ⇔ (iii) Follows from Proposition 2.3.(vi) ⇔ (i) Follows from section 4.2.(vii) ⇔ (i) Follows from section 4.5.(i) ⇔ (viii) Follows from section 4.4 and the uniqueness ofX.
This finishes the proof of the Main Theorem.
6.2. Remark. If we take a sublatticeA2 ⊥ A2 in U ⊥ Λ, its orthogonalcomplement is isomorphic toU ⊥ E8 ⊥ E6 ⊥ E6. This lattice is isometricto the Picard lattice of the supersingular K3 surface in characteristic 3 withthe Artin invariant 1. It is known that this K3 surface is isomorphic to theFermat quartic surface ([Sh2], Example 5.2). By using the same method asin this paper we can see that the projection of the Weyl vectoris the class ofa hyperplane section of the Fermat quartic surface, 112 lines on the Fermatquartic surface can be written in terms of Leech roots and theprojectiveautomorphism group PGU(4, F3) of the Fermat quartic surface appears as asubgroup of Aut(D).
20 I. DOLGACHEV AND S. KONDO
6.3. Remark. A lattice is calledreflectiveif its reflection subgroup is offinite index in the orthogonal group. The Picard latticeU ⊥ D20 is reflec-tive. This was first pointed out by Borcherds [Bo] and it is theonly knownexample (up to scaling) of an even reflective lattice of signature(1, 21).
Esselmann [Es] determined the range of possible ranksr of even reflec-tive lattices of signature(1, r − 1): 1 ≤ r ≤ 20 or r = 22. This is the sameas the possible ranks of the Picard lattices of K3 surfaces. Of course, therankr = 22 occurs only when the characteristic is positive.
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[Coo] J. L. Coolidge,A Treatise on Algebraic Plane Curves, Dover Publ., New York,1959.
[Co1] J. H. Conway,Three lectures on exceptional groups, in Finite simple group, pp.215–247, Academic Press, New York 1971 (see also J. H. Conway, N. J. A.Sloane,Sphere packings, lattices and groups, Chap. 10, Grundlehren Math. Wiss.Bd 290, 3rd ed., Springer-Verlag, Berlin, Heidelberg, New York 1999).
[Co2] J. H. Conway,The automorphism group of the26 dimensional even Lorentzianlattice, J. Algebra80 (1983), 159–163 (see also J. H. Conway, N. J. A. Sloane,Sphere packings, lattices and groups, Chap. 27, Grundlehren Math. Wiss. Bd 290,3rd ed., Springer-Verlag, Berlin, Heidelberg, New York 1999).
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[Mu] S. Mukai, Finite automorphism groups of K3 surfaces and the Mathieu group,Invent. Math. 94 (1988), 183-221.
[RS1] A. Rudakov, I. Shafarevich,Supersingular K3 surfaces over fields of character-istic 2, Izv. Akad. Nauk SSSR, Ser. Mat. 42, (1978), 848–869 (see also Shafare-vich’s ”Collected Mathematical Papers”, Springer-Verlag, 1989, pp.614–632).
[RS2] A. Rudakov, I. Shafarevich,Surfaces of type K3 over fields of finite characteristic,Itogi Nauki Tekh, Ser. Sovrem Problems Mat. 18 (1981), 115–207 (see also Sha-farevich’s ”Collected Mathematical Papers”, Springer-Verlag, 1989, pp.657–714).
[Sh1] T. Shioda,On elliptic modular surfaces, J. Math. Soc. Japan24 (1972), 20–59.[Sh2] T. Shioda,SupersingularK3 surfaces, Algebraic Geometry (Proc. Summer Meet-
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A SUPERSINGULAR K3 SURFACE AND THE LEECH LATTICE 21
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF M ICHIGAN , ANN ARBOR, MI48109,USA
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