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We consider ¢rst the representation theory of H10 on the space of quadrics H0_O

P9…2††. There are four types of ¢ve-dimensional irreducible representations

appearing in H0_O

P9…2††. Indeed, we can write H0…O_P9…2†† 3V₁3V2

3V32V4;. The three representations of type V1 are given by the span of f,

s…f†;. . .;s4_{f†, where f ˆx}2

0‡x25, x1x9‡x6x4, or x2x8‡x7x3. Three represen- tations of type V2 are given similarly with f ˆx0x5, x9x6‡x4x1, or x8x7‡x3x2. Three representations of type V3 are given by f ˆx20ÿx52, x1x9ÿx6x4 or

x2x8ÿx7x3, and ¢nally the two representations of type V4 are given by f ˆx9x6ÿx4x1 or x8x7ÿx3x2.

Note that for y2P3

ÿ, the matrixM5…x;y† is skew-symmetric, while for y2P5‡,

M5…x;y† is symmetric. For a general parameter point y2P3ÿ, the 44 Pfaf¢ans

of M5…x;y† cut out a variety GyP9 which can be identi¢ed with a PlÏcker

embedding of Gr(2,5) in P9_{. The corresponding varieties de¢ned by the Pfaf¢ans} of M5…s5_{x†;y†, M5…s}5_t5_{x†;y† and M5…t}5_{x†;y† are then s}5_G

y†, s5t5…Gy† and

t5_G

y†, respectively.

Note that the subgrouphs5_;t5_iofH

10acts onP3ÿ, and we denote the quotient of

this action by P3

ÿ=Z2Z2. By [GP1], Theorem 6.2, we have a map

Y10:Alev…1;10†ÿ!P3ÿ=Z2Z2

which essentially maps an Abelian surface to the orbit of its odd 2-torsion points. The mapY10is birational onto its image. Since both spaces are three dimensional, we obtain the following result:

THEOREM 7.1. Alev

10 is birationally equivalent toP3ÿ=Z2Z2. & Remark7.2. The quotientP3

ÿ=Z2Z2, being a quotient ofP3by a ¢nite Abelian group, is rational by [Miya]. One can also compute explicitly the ring of invariants of this action, and this ring is generated by quadrics and quartics. There are 11 inde- pendent invariants of degree 4, and these can be used to map P3

ÿ=Z2Z2intoP10,

where one ¢nds the image to be a singular Fano threefold of index 1 and genus 9. We also have

THEOREM 7.3. Let y2P3

ÿ be a general point. Then

(1) The 20 quadrics de¢ning the four PlÏcker embedded Grassmannians Gy,s5…Gy†,

s5_t5_G

y†, t5…Gy† span only a 15-dimensional space of quadrics

RyH0…OP9…2††, which as a representation of H₁₀ is of type V₁V₂V₃;

(2) The subspace Rygenerates the homogeneous ideal of the(1,10)-polarized Abelian

surface corresponding to the image via Y10 of y in P3ÿ=Z2Z2.

Proof. By [GP1], Corollary 2.7, ifAis aH10invariant Abelian surface inP9and y2A\P3

ÿ is an odd 2-torsion point, then AGy\s5…Gy† \t5…Gy† \s5t5…Gy†,

so all quadrics in the subspaceRyvanish onA. In fact, Theorem 6.2 of [GP1] states

that AˆGy\s5…Gy† \t5…Gy†, and, moreover, the corresponding space of 15

quadrics generates the homogeneous ideal of A in P9_{. Thus we must have dim} Ryˆ15. To see howRydecomposes as anH10representation, it is suf¢cient to check the isomorphismRyV1V2V3for a special value ofy. One may use the point yˆ …0:1:1:0:0:0:0:0:ÿ1:ÿ1† 2P3

ÿ, which in the notation of [GP1], }4, yields

the degenerate surface X1

generated by the quadrics

fxixi‡2‡xiÿ1xi‡3ji2Z10g [ fxixi‡5ji2Z10g:

This clearly decompose as V1V2V3. &

THEOREM 7.4. (1)For general y2P3

ÿ,V10;yˆGy\t5…Gy† P9 is a Calabi^Yau

threefold with50ordinary double points, which is invariant under the subgroup gen- erated by s2 _{and t. The singular locus of V}

10;y is the hs2;ti-orbit of fs5…y†g. If

AV10;y is the H10-invariant (1,10)-polarized Abelian surface which has y as an odd 2-torsion point, then the linear system j2HÿAj induces a 2 : 1 cover from V10;y to the symmetric HM-quintic threefold X5;y0 P4, where

y0_{ˆ …y}

1y2‡y3y4:ÿy2y3:y1y4:y1y4:ÿy2y3† 2P2‡P4

(see Proposition 3.4). (2)If y2A\P5

‡,let W10;yP9 be the variety de¢ned by the 33 minors of the

matrix M5…x;y†. For general y, this is a Calabi^Yau threefold of degree 35 with 25 ordinary double points. Furthermore, W10;y is not simply connected, but has

an unbranched double cover birational to X_5;y00 P4,with

y00_{ˆ …2…y}

3y4ÿy1y2†:y0y1ÿy4y5:y2y5ÿy0y3:y2y5ÿy0y3:y0y1ÿy4y5† 2P2

‡P4:

Proof: (1) To understand the singularity structure ofV10;y, we will ¢rst understand

the image of the map induced by the linear systemj2HÿAjinP4_{, and show that this} map expressesV10;yas a partial resolution of a double cover ofX5;y0 P4branched

over its singular locus.

First we note that for generaly2P3

ÿ,V10;yis an irreducible threefold of degree 25.

To see this, we show it is true for special choice ofy. Choosey2P3

ÿso thatyiˆy5‡i.

ThenM5…x;y† ˆ …x0

i‡jyiÿj†wherex0i‡j ˆxi‡j‡xi‡j‡5. ThusGyis just a cone over an

elliptic normal curve Ein the P4 _{given byfx}

iÿxi‡5ˆ0g, with vertex theP4 given by fxi‡xi‡5ˆ0g. Gt5_y† is a similar cone over t5…E†, and then V_10;y ˆG_y\G_t5_y†

is just the linear join of E and t5_{E†, which is an irreducible threefold of degree} 25, by Proposition 1.4. For general y,Gy andGt5_y† have dimension 6 and degree

5 in P9_{, so G}

y\Gt5_y† is expected to be of dimension 3 and degree 25. Since this

holds for special y, it holds for generaly.

Next we show that V10;y contains a pencil of Abelian surfaces. Letl be the line

joining y and t5_{y†, so that s}5_{l† is the line joining s}5_{y† and t}5_s5_{y†. Let} z2s5_{l†, such that z2 f= s}5_{y† ‡s}5_t5_y†;s5_{y† ÿs}5_t5_{y†g. Then there is a linear} transformation T2PGL(P9_{) of the form Tˆdiag(a,b,. . .,a,b) such that} T2_{z† ˆs}5_{y†. Note that t}5_TˆTt5_{, s}5_{T ˆT}ÿ1_s5 _{in PGL(P}9_{), and that for any} w2P3

ÿ, we haveT…Gw† ˆGTÿ1_w†. (The signi¢cance of these communtation relations

surface determined by T…z† 2P3 ÿ, i.e. AT…z†ˆGT…z†\Gs5_T…z†\G_t5_T…z†\G_s5_t5_T…z†: Then T…AT…z†† ˆGz\Gs5_T2_z†\G_t5_z†\G_s5_t5_T2_z† ˆGz\Gt5_z†\ …G_y\G_t5_y†† V10;y:

Aszvaries ons5_{l†, the pointT(z) also varies on this line, and thus we obtain a pencil} of distinct (1,10)-polarized Abelian surfaces inV10;y, ¢lling upV10;yby irreducibility.

Note that in particular,s5_{z† ˆT…s}5_{T…z††† 2T…A}

T…z††moves along the linel. Thus

lV10;y, and each Abelian surface in the pencil intersects the linel.

Now we will study the linear system j2HÿAj on V10;y. To do so, we must ¢rst

understand the equations ofV10;ymore deeply. The Pfaf¢an of the matrix obtained

by deleting the ¢rst row and column of M5…x;y† is

pˆ ÿx0x5…y1y2‡y3y4† ‡ …x1x4‡x6x9†y2y3ÿ …x2x3‡x7x8†y1y4‡ ‡ …x3x7y21‡x2x8y24† ÿ …x4x6y22‡x1x9y23† ‡ …x25y1y3‡x20y2y4†: The ideal of V10;y is then generated by

fs2i_{p† j0WiW4g [ ft}5_s2i_{p† j0WiW4g:}

On the other hand, the ideal of A is generated by these ten quadrics plus the additional quadrics fs2i‡5_{p† j0WiW4g. Thus in particular, if we set}

fi:ˆ1₂s2i…s5…p‡t5…p†† ‡p‡t5…p††

we see that

fiˆ …x3‡2ix7‡2i‡x2‡2ix8‡2i†…y21‡y24†‡ ‡ …x2

5‡2i‡x22i†…y1y3‡y2y4† ÿ …x4‡2ix6‡2ix1‡2ix9‡2i†…y22‡y23†;

and thatf0;. . .;f4are linearly independent quadrics which are still linearly indepen- dent when restricted to V10;y. Furthermore, each fi vanishes on A, and in fact

f0;. . .;f4 cut outA on V10;y. Thus f0;. . .;f4 de¢ne a four-dimensional linear sub- system of jH0_OV

10;y…2††j whose base locus is precisely the Abelian surfaceA.

We now usef0;. . .;f4to de¢ne a rational mapf:V10;y- - P4. We describe next its

image. The mapfis induced by the linear systemj2HÿAjonV2

10;y, where the model

V2

10;yÿ!V10;y is obtained by blowing up V10;y along A. We also denote by f the

induced morphism V2

10;yÿ!P4.

Letzbe a general point on the lines5_l†joinings5_y†ands5_t5_{y†, andTas before.} Let us consider the restriction of f to the surface A0_ˆT…A

T…z††. There are two

alternatives. First if A\A0 _{is an isolated set of points, then the number of these}

points is divisible by 50, since A and A0 _{are invariant under the action of tand}

s2_{. Sinces}5_{y† 2A\A}0_{, we have at least 50 such points, and the degree off:A}0_{- - P}4 is 4 20ÿa 50, for someaX1, which is nonnegative only ifaˆ1. ThusA\A0_can

only consist of onehs2_{;ti-orbit ofs}5_{y†, andAmust intersectA}0_{transversally. Thus}

the proper transform of A0 _{in V}2

10;y, which we denote by eA0, is A0 blown up in

50 points.

The second alternative is that A\A0_{contains a curveC. In this case, since Ais}

general, the curve C can only be numerically equivalent to nH on A0_{, for some}

n. Since A is cut out by quadrics, nW2. On the other hand, since A and A0 _are

invariant with respect to s2 _{andt, so isC. However, by [LB], Ex. (4), p. 179, this} is impossible. Thus this case does not occur!

Next observe that forx2V10;y,f…x† ˆf…t5…x††, since eachfiist5-invariant. Thusf

factors viaV10;yÿ!V10;y=ht5i- - P4. Nowt5acts onA0. The (1,10) polarizationLon

A0_{descends to a (1,5) polarizationL}0_onA0_=t5_{, andf}

0;. . .;f4descend to sections of L02. The map f: A0_=t5_{- - P}4 _{lifts then to a morphism f~: A~}0_=t5_ÿ!P4_{. It is easy} to see that f~ satis¢es all the hypotheses of Proposition 3.6. Thus f maps

T…AT…z††, forzgeneral, to a Heisenberg invariant Abelian surfaceA00P4of degree

15, and A00 _{is contained in a quintic X5;y0P}4 _{for any y}0_2A00_{. If we ¢nd a point}

y0_{which is contained in f…T…A}

T…z†††for allz2s5…l†, thenf…V10;y† X5;y0. A simple

calculation shows that

f…l† ˆ …y1y2‡y3y4:ÿy2y3:y1y4:y1y4:ÿy2y3† ˆ:y0:

SinceT…AT…z††intersectslfor eachz,y0is the desired point, and sof…V10;y† X5;y0. To

show thatfmaps two-to-one ontoX5;y0, we observe that onV₁₀2_;y,HandAare Cartier

divisors, with H3 _{ˆ25, H}2_Aˆ20,HA2_{ˆ0 andA}3_{ˆ ÿ50, (the latter holds since} …2HÿA†2_{Aˆ30, asf maps A two-to-one to a degree 15 surface in P}4_{). From} this we see that …2HÿA†3_{ˆ10. Thus f…V}

10;y† ˆX5;y0 and f maps two-to-one

generically.

To compute the branch locus of the double cover f: V2

10;yÿ!X5;y0, we need to

identify the ¢xed locus of the involution t5 _{acting on V}

10;y. This is easily done:

the ¢xed locus of t5 _{acting on P}9 _{consists of} L0ˆ fx0ˆx2ˆx4 ˆx6 ˆx8ˆ0g and

L1ˆ fx1ˆx3ˆx5 ˆx7 ˆx9ˆ0g: Note we can write

M5…x;y† ˆ …x6…i‡j†y6…iÿj†‡x6…i‡j†‡5y6…iÿj†‡5†0Wi;jW4; making it clear that

M5…x;y†jL0 ˆ ÿM5…t5…x†;y†jL0 ˆ …x6…i‡j†‡5y6…iÿj†‡5†0Wi;jW4

"

"

and

M5…x;y†jL1 ˆM5…t5…x†;y†jL1ˆ …x6…i‡j†y6…iÿj††0Wi;jW4:

Each of these are skew-symmetric Moore matrices on P4_{. Thus by [Hu1] and} [ADHPR2], Proposition 4.2, each of these matrices drops rank on an elliptic normal curve. Thus …L0[L1† \V10;yˆE0[E1, the disjoint union of two elliptic normal curves. This is the ¢xed locus of t5 _{acting on V}

10;y. It is now clear that f must

map E0 andE1 to the two singular curves of X5;y0 of Proposition 3.4, (3).

We can now understand from this the singularity structure ofV2

10;y. First note that

the exceptional locus ofp2:V102;yÿ!V10;yconsists of 50P1's.Ais a Cartier divisor on

V2

10;y, so in particularV102;yis nonsingular along any nonsingular element of the linear

systemjAj. But the exceptional locus ofp2is contained in the base locus ofjAj, hence V2

10;y is nonsingular along the exceptional locus. In particular, sincep2 is the small resolution of 50 ordinary double points and oV10;y is trivial, so isoV₁₀2_;y.

LetV2 10;yÿ!

s1 _^

X5;y0ÿ!s2 X_5;y0, be the Stein factorization off:V₁₀2_;yÿ!X_5;y0. Thens₂

is branched precisely over Sing…X5;y0†, and is a double cover. From the description of

the singularities of X5;y0 of Proposition 3.4, (5), one sees thats2…Sing…X^_5;y0††is the

Heisenberg orbit of y0_{, and shows that X^}

5;y0 has one ordinary double point sitting

over each point of the orbit ofy0_.

Next considers1:V102;yÿ!X^5;y0. We have already seen thatf(l)ˆy0; thuss₁ must

contract l and its hs2_{;ti-orbit (25 lines) to the 25 ordinary double points of} ^

X5;y0. Sinceo_V2

10;y andoX^5;y0 are trivial,s1 must be crepant, and the only possibility

then is that s1 only contracts these 25 lines, while V102;y is nonsingular. Thus

V10;y itself has 50 ordinary double points obtained by contracting the exceptional

locus of p2.

(2) Fix any point y2P5

‡. Consider ¢rst P…Sym2…C5†† ˆProjC‰fxijj0Wi;jW4;

xijˆxjigŠ. The generic symmetric matrix Mˆ …xij† has rank 2 precisely on

Sym2_P4_{† P…Sym}2_C5_{††. One computes that Sym}2_P4_{† has degree 35 in this} embedding and thus W10;y can be described as i…P9† \Sym2…P4†, where i:

P9_ÿ!P…Sym2_C5_{†† is the linear map given by x}

ijˆxi‡jyiÿj‡xi‡j‡5yiÿj‡5. Thus, if the intersection is transversal, W10;y has dimension 3 and degree 35.

To understandW10;yfurther, consider it embedded inSym2…P4†via the mapi. Let

p: P4_P4_ÿ!Sym2_P4_{† be the quotient map, and let z}

0;. . .;z4;w0;. . .;w4 be coordinates on the ¢rst and second P4_{'s, respectively. The map p is given by} xijˆziwj‡zjwi. Let us determine the equations of pÿ1…W10;y†, which is a double

cover of W10;y. One checks easily that the equations of i…P9† are

Hence pÿ1_W

10;y†is given by the ¢ve bilinear equations

f…y3y4ÿy1y2†…2ziwi† ‡ …y0y1ÿy4y5†…ziÿ1wi‡1‡zi‡1wiÿ1†‡ ‡ …y2y5ÿy0y3†…ziÿ2wi‡2‡zi‡2wiÿ2† ˆ0g:

If we write

y00

0ˆ2…y3y4ÿy1y2†; y001ˆy004ˆy0y1ÿy4y5; y002ˆy003ˆy2y5ÿy0y3; then we can rewrite the above set of equations asL(z,y00_{)wˆ0, whereL(z,y}00_{) is the}

55 matrix L…z;y00_{† ˆ …z}

2iÿjy00iÿj†. If p1;p2: P4P4ÿ!P4 are the ¢rst and second projections, then X0

5;y00 ˆp1…pÿ1…W10;y†† is given by the equationfdet L(z,y00)ˆ0g.

Thus it is clear that W10;yis of dimension three and hence of degree 35. Note that

this argument shows that a generic intersection ofSym2…P4_†andP9_{is a nonsingular} threefold (since the singular locus of Sym2_P4_{† has codimension 4) and has an}

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