On Weddle Surfaces And Their Moduli

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On Weddle Surfaces And Their Moduli

Michele Bolognesi

To cite this version:

Michele Bolognesi. On Weddle Surfaces And Their Moduli. 2006. �hal-00016821v3�

ccsd-00016821, version 3 - 4 Oct 2006

Mihele Bolognesi

Abstrat

TheWeddlesurfaeislassiallyknown tobeabirational (partiallydesingularized)modelof

the Kummer surfae. Inthis note wego throughits relations with moduli spaesof abelian

varietiesandofranktwovetorbundlesonagenus2urve. Firstweonstrutamodulispae

A

₂

(3)

⁻ parametrizing abelian surfaes with a symmetri theta struture and an odd theta

harateristi. Suhobjetsanin fat beseenasWeddlesurfaes. Weprovethat

A

₂

(3)

^{− is}

rational. Then,givenagenus2urve

C

,wegiveaninterpretationoftheWeddle surfaeasa modulispaeofextensionslasses(invariantwithrespettothehyperelliptiinvolution)ofthe

anonialsheaf

ω

of

C

with

ω

⁻¹. ThisinturnallowstoseetheWeddlesurfaeasahyperplane setionoftheseantvariety

Sec ( C )

^oftheurve

C

trianoniallyembeddedin

P

4

.

Introdution

The Burkhardt quarti hypersurfae

B ⊂ P

⁴ is a hypersurfae dened by the vanishing of theunique

Sp(4, Z /3 Z )/ ± Id

^{invariantquarti}polynomial. Itsexpliitequationwaswritten downforthersttimebyH.Burkhardtin1892 [Bur92℄. ItwasprobablyknowntoCoble(or

atleastone an inferthatfromhisresults)thatageneripointof

B

^{representsa prinipally}

polarizedabeliansurfae(ppasforshort)withalevel3struturebutitwasonlyreentlythat

G.VanderGeer [vdG87℄madethisstatementlearer. InpartiularVander Geer([vdG87℄,

Remark 1) pointed out the fat that the Hessian variety

Hess(B)

^{of the Burkhardt quarti}

is birationalto the modulispae parametrizing ppas with a symmetri theta struture and

an even theta harateristi, whih we will denote by

A

₂

(3)

^{+. The modulispae}

A

₂

(3)

^{+ is}

onstrutedasaquotientofthe Siegelupperhalfspae

H

₂ by the arithmetigroup

Γ

2

(3, 6)

^.

Moreover, sine

B

^is self-Steinerian ([Hun96℄, Chapter 5),one an viewthe 10:1 Steinerian map

St

+

: Hess(B) −→ B

⁽¹⁾

as the forgetful morphism

f : A

₂

(3)

⁺

→ A

₂

(3)

^{whih forgets the symmetri line bundle}

representing the polarization. This means that the following diagram,where the horizontal

arrows

T h

^{+ and}

Q

^{are birational}isomorphisms,ommutes.

A

₂

(3)

⁺

−→

^{T h+}

Hess(B) ⊂ P

⁴

f ↓ ↓ St

₊

A

₂

(3) −→

^Q

St

+

(B) = B

Coble alsoomputed indetail a unirationalization

π : P

³

−→ B,

with the Steinerian map (1), the degree of this map has lead us to suspet that

P

³ ould bebirationaltoanother modulispae,whihwedenoteby

A

2

(3)

^{−,that should}parametrize ppas with a symmetri theta struture and an odd theta harateristi. In this paper we

desribe the arithmeti group

Γ

2

(3)

^{− whih realizes}

A

2

(3)

^{− asa quotient}

A

₂

(3)

⁻

= H

₂

/Γ

2

(3)

⁻

.

Moreover we prove the following theorem.

Theorem 0.0.1 Let

A

₂

(3)

^{− be the moduli spae of ppas with a symmetri level 3 struture}

and an odd theta harateristi. The theta-null map

T h

^{− given by even theta funtions}

indues a birationalisomorphism

T h

⁻

: A

₂

(3)

⁻

−→ P

³

.

Furthermore, the pullbak by

π

^{of tangent hyperplane setions of}

B

^{are Weddle quarti}

surfaes. Let

C

^{be a genus 2 urve and}

τ : ξ 7→ ξ

⁻¹

⊗ ω

^{the Serre involution on the Piard}

variety

P ic

¹

(C)

^{. Chosen an} appropriate linearization for the ation of

τ

^on

O

P ic¹(C)

(Θ)

^,

the Weddle surfae

W

^{is the image of}

P ic

¹

(C)

ⁱⁿ

P

³

= P H

⁰

(P ic

¹

(C), 3Θ)

^∗₊ ^{(where the plus}

indiatesthatwe areonsideringinvariantsetions). Moreoverthe surfae

W

^{isabirational}

modelof the Kummer surfae

K

¹

= P ic

¹

(C)/τ ⊂ P H

⁰

(P ic

¹

, 2Θ)

^{∗. Given appas}

A

^withan

odd linebundle

L

representing the polarization(resp. aneven linebundle) one an aswell obtain a Weddle surfae by sending

A

^{in the}

P

³ obtained from the eigenspae

H

⁰

(A, L

³

)

+

(resp.

H

⁰

(A, L

³

)

−^{) w.r.t. the standard involution}

±Id

^{. Sine also this Weddle surfae is a}

birational model of the Kummersurfae

K := A/ ± Id ⊂ P H

⁰

(A, L

²

)

^{∗, we go through the}

onstrution of the birational map between the two surfaes, proving that it omes (in the

odd linebundlease) from aanonial embedding

Q : H

⁰

(A, L

²

)

^∗

֒ → Sym

²

H

⁰

(A, L

³

)

+

.

⁽²⁾

Furthermore, apoint of

A

₂

(6)

^{an be assoiatedto suha ongurationof surfaes.}

Inthe seond(independent)part ofthe paperwehangeour pointof view: wex asmooth

genus 2 urve

C

^{and onsider the moduli spae}

M

_C ^{of ranktwo vetor bundles on}

C

^with

trivial determinant. It is well known [NR69℄ that

M

C ^{is isomorphi to}

P

³, seen as the

2Θ

^-

linear serieson the Jaobianof

C

^{and that the semistableboundaryis the Kummersurfae}

K

⁰

= Jac(C)/ ± Id ⊂ |2Θ|

^{. The spae}

P Ext

¹

(ω, ω

⁻¹

) ∼ = P

⁴

= |ω

³

|

^∗ parametrizesextensions lasses

(e)

^of

ω

^by

ω

^−1.

0 −→ ω

⁻¹

−→ E

_e

−→ ω −→ 0. (e)

One hosen appropriate ompatible linearizations on

P ic

¹

(C)

^and

C

^{, we show that the}

linear system

P H

⁰

(P ic

¹

(C), 3Θ)

^∗₊ ^{an be injeted in}

P Ext

¹

(ω, ω

⁻¹

)

^{and that we have the}

followingtheorem.

volutioninvariantextensionlassesof

ω

^by

ω

^{−1 istheWeddlesurfae}

W ⊂ P H

⁰

(P ic

¹

(C), 3Θ)

^∗₊

assoiated to

P ic

¹

(C)

^.

Moreover, let

Sec(C) ⊂ |ω

³

|

^{∗ be the seant varietyof the urve}

C

trianonially embedded, we show that

W

^{is the} (everywhere tangent) intersetion of

Sec(C)

^{with the hyperplane}

given by

P H

⁰

(P ic

¹

(C), 3Θ)

^∗₊^.

Aknowledgments. It is a pleasure to thank my thesis advisor Christian Pauly, without

whose insight and suggestions this paper ouldn't have been written. I'm alsovery grateful

to Bert Van Geemen for the inuene he has had on my formation and the passion he has

transmitted me.

1 Theta harateristis and ongruene subgroups of

Sp(4, Z )

1.1 Theta harateristis

Formuhof the materialinthis setionthe refereneis[Bea91℄. Let

(A, H)

^{bea prinipally}

polarized abelian variety (ppav for short) of dimension

g

^{. We willdenote}

A[2]

^{the group of}

2-torsionpointsand let

h , i : A[2] × A[2] → {±1}

be the sympleti formindued by the prinipalpolarization.

Atheta harateristi of

A

^{isaquadratiform}

κ : A[2] → {±1}

^{assoiatedtothesympleti}

form

h , i

^{, i.e. a funtionon}

A[2]

^verifying

κ(x + y)κ(x)κ(y) = hx, yi,

forevery

x, y ∈ A[2]

^{. Wewilldenotetheset oftheta}harateristisby

ϑ(A)

^{. Let}

x, y ∈ A[2]

and

κ ∈ ϑ(A)

^{. The}

F

₂-vetor spae

A[2]

^{ats on}

ϑ(A)

^{inthe following way}

(x · κ)(y) = hx, yiκ(y)

and

ϑ(A)

^{is an}

A[2]

^{-torsor w.r.t. this ation. Let}

κ

^{be an element of}

ϑ(A)

^{, there exists a}

number

ǫ(κ) ∈ {±1}

^s.t.

κ

^{takes the value}

+ǫ(κ)

^(resp.

−ǫ(κ)

^{) at}

2

^g−1

(2

^g

+ 1)

^points(resp.

2

^g−1

(2

^g

− 1)

^{points). The theta} harateristi is said to be even if

ǫ(κ) = +1

^{, odd in the}

opposite ase, we willwrite

ϑ

⁺

(A)

^and

ϑ

⁻

(A)

^{forthe two sets just dened. Given}

x ∈ A[2]

^,

ǫ

^satises

ǫ(x · κ) = κ(x)ǫ(κ).

⁽³⁾

Let

T (A)

^bethe

A[2]

^{-torsorof symmetritheta divisors}representing the polarization,there is aanonial identiationof

A[2]

^{-torsors(whih wewillimpliitly make inwhat follows)}

ϑ(A) −→

^∼

T (A),

⁽⁴⁾

κ 7→ Θ

κ

.

This sends athetaharateristi

κ

^of

A

^{toasymmetri thetadivisor}

Θ

κ ^on

A

haraterized by the formula

κ(a) = (−1)

^{ma(Θκ)+m0(Θκ)}

,

where

a ∈ A[2]

^and

m

a

(Θ

κ

)

^{is the} multipliity of the divisor

Θ

κ ^{at the point}

a

^{. Let}

a ∈ A

and

t

a ^{be the} translation

x 7→ x + a

ⁱⁿ

A

^{, then}

Θ

a·κ

= t

^∗_a

Θ

κ ^and

ǫ(κ) = (−1)

^m0(Θκ)

.

Thusthe fat thata thetaharateristi iseven orodd depends onthe loalequation of

Θ

κ

atthe origin.

Remark 1.1.1 Suppose

A = Jac(C)

^{is the Jaobian of a urve}

C

^{, and denote by}

ϑ(C) ⊂ P ic

^{g−1 thesetofthetheta}harateristis of

C

^{,i.e. linebundles}

L

^s.t.

L

²

= ω

^{. Then}

ϑ(C) ∼ = ϑ(Jac(C)) ∼ = T (Jac(C))

^as

Jac(C)[2]

^-torsorsby

L 7→ Θ

L

= {M ∈ Jac(C)|H

⁰

(L ⊗M ) 6= 0}

^;

and

ǫ

^{is the usual parity funtion bythe Riemann} singularity theorem.

Letusdenoteby

ı

^{theinvolution}

−Id

^ontheppav

A

^{. Let}

θ

_κ ^{beanonzerosetionof}

O

_A

(Θ

_κ

)

^,

and

φ

^{the unique} isomorphism between

ı

^∗

O

A

(Θ

κ

)

^and

O

A

(Θ

κ

)

^{whih indues the identity}

over the origin. Following Mumford [Mum66℄, we will all

φ

^{the normalized} isomorphism.

Then we have

φ(ı

^∗

θ

κ

) = ǫ(κ)θ

κ

.

⁽⁵⁾

Denition 1.1.2 Let L be a symmetri line bundle representing the polarization H, let

φ : L → ı

^∗

L

^{be the normalized} isomorphism and

x ∈ A[2]

^{. We dene}

e

^L_∗

(x)

^{as the salar}

α

^s.t.

φ(x) : L

x

→

∼

(ı

^∗

L)

x

= L

ı(x)

= L

x

is the multipliation by

α

^.

The funtionthatassoiates thesalar

e

^L_∗

(x)

^toapoint

x ∈ A[2]

^{isa quadratiformon}

A[2]

and, if

κ ∈ ϑ(A)

^then

e

^O(Θ∗ ^{κ) is the quadrati form}

κ

^{[Mum66℄. We willoften say thata line}

bundle iseven (resp. odd) if the indued quadrati formon

A[2]

^{is even (resp. odd).}

Any given

κ ∈ ϑ(A)

^{an be used toidentify}

ϑ(A)

^with

A[2]

^{,via the} isomorphism

A[2] −→

^∼

ϑ(A),

⁽⁶⁾

x 7→ x · κ.

1.2 Moduli spaes and subgroups of

Sp(2g, Z )

Let

g

^{be a positive integer and}

Γ

g

= Sp(2g, Z )

^{the full Siegel modular group of genus}

g

^.

When neessary, wewillusefor the elements

M ∈ Γ

g ^{the usual} deompositioninfour

g × g

^-

bloks,

M =

A B C D

and if

Z

^{isa squarematrix, we willwrite}

Z

^{t foritstranspose. The}

group

Γ

_g ^{ats properly}disontinuously and holomorphiallyon the Siegelupper half-plane

H

_g

:= {Ω ∈ Mat

g

( C )|Ω = Ω

^t

, Im(Ω) > 0}

by the formula

M · Ω = (AΩ + B)(CΩ + D)

⁻¹

.

⁽⁷⁾

The quotient

A

_g

:= H

_g

/Γ

g ^{is a} quasi-projetive variety and it an be seen as the oarse modulispae of ppav of dimension

g

^{[Igu72℄. Let}

m

^{bea vetor of}

(

¹₂

Z / Z )

^{2g. Suh avetor}

isusuallyalledahalf-integerharateristiandwewillall

a

^and

b

^therstandrespetively the seond

g

-oordinates of

m

^{. One we hoose a}

Ω ∈ H

_g, we an assoiate to every half- integer harateristi a holomorphi theta funtion on the abelian variety orresponding to

Ω

^mod

Γ

g ^{as follows}

Θ a

b

( z ; Ω) := X

r∈Z^g

e

^{πi((r+12a)·Ω·(r+12a)+2(z+12b)·(r+12a))}

.

Moreover the zero divisor of

Θ a

b

is a symmetri theta divisor. Thus, via the identia-

tion 4, one an dene (although non anonially) bijetions between the set of half-integer

harateristisand

ϑ(A)

^{. F}urthermore,the ationof

Γ

g ^on

Ω ∈ H

_g induesatransformation formula for theta funtionswith harateristis([Igu64℄, Setion 2). The indued ationon

the harateristis is then the following

M · a

b

=

D −C

−B A

a b

+ 1

2

diag(CD

^t

) diag(AB

^t

)

.

⁽⁸⁾

Lemma 1.2.1 ([Igu64℄, Setion 2)

The ation of

Γ

g ^on

(

¹₂

Z / Z )

^{2g dened by (8) has two orbits} distinguished by the invariant

e (m) = (−1)

^4abt

∈ {±1}.

We say that

m

^{is an even (resp. odd)} half-integerharateristi if

e (m) = 1

^(resp.

e (m) =

−1

^{)and this invariant oinides via(4)with the invariant}

ǫ

^{dened ontheta} harateristis in Setion1. Let usdenote by

Γ

g

(3) := Ker(Sp(2g, Z ) → Sp(2g, Z /3 Z ))

the prinipal ongruene group of level 3 and by

Γ

g

(3, 6)

^{the subgroup of}

Γ

g

(3)

^{dened by}

diag(CD

^t

) ≡ diag(AB

^t

) ≡ 0 mod 6

^{. Thesubgroup}

Γ

g

(3, 6)

^{thenoinideswiththestabilizer}

of the even theta harateristi

0 0

.

2 Symmetri theta strutures

Let

(A, H)

^{bea ppav of dimension}

g

^{and let}

L

^{bea symmetrilinebundlethat indues the}

polarization on it. Let

z ∈ A

^and

t

_z ^{be the} translation

x 7→ x + z

^on

A

^{. The level 3 (and}

genusg) theta group of

L

^{is dened in the followingway}

G(L

³

) = {(ϕ, η)|η ∈ A, ϕ : t

^∗_η

(L

³

) →

^∼

(L

³

)},

where the group lawis

(ϕ, η) · (ϕ

^′

, η

^′

) = (t

^∗_η′

ϕ ◦ ϕ

^′

, η + η

^′

)

^.

Group theoretially one an see

G(L

³

)

^{asa entralextension}

1 −→ C

^∗

−→ G(L

^{i 3}

) −→

^p

A[3] −→ 1,

where the image of

α

^via

i

^isthe automorphismof

L

^{3 given by the} multipliationby

α

^and

p(ϕ, η) = η

^{. Theommutator}

[(ϕ, η), (ϕ

^′

, η

^′

)]

^{oftwoelementsof}

G(L

³

)

^{belongsto the enter}

of the group and it induesthe Weil pairing

e

^L

: A[3] × A[3] → C

^∗

taking lifts. Two dierent lifts give the same ommutator.

As anabstrat group

G(L

³

)

^{isisomorphi to the Heisenberg group}

H

_g

(3) := C

^∗

× ( Z /3 Z )

^g

× ( Z [ /3 Z )

^g

,

where

( Z [ /3 Z )

^g

:= Hom(( Z /3 Z )

^g

, C

^∗

)

^{. The group law in}

H

g

(3)

^{is not the produt law but}

the following

(t, x, x

^∗

) · (s, y, y

^∗

) = (stω

^y∗(x)

, x + y, x

^∗

+ y

^∗

),

where

ω

^{is a ubi root of 1. The projetion}

(t, x, x

^∗

) 7→ (x, x

^∗

)

^{denes a entral extension}

of groups

1 −→ C

^∗

−→ H

_g

(3) −→ ( Z /3 Z )

^2g

−→ 1.

Let

u := (x, x

^∗

), v := (y, y

^∗

) ∈ ( Z /3 Z )

^{2g and}

u, ˜ v ˜ ∈ H

_g

(3)

^{two lifts. Then the ommutator}

[˜ u, ˜ v]

^{doesnot depend onthe hoie of the lifts and it denes the standard sympletiform}

E

^on

( Z /3 Z )

^{2g, that is}

E : ( Z /3 Z )

^2g

× ( Z /3 Z )

^2g

−→ C

^∗

;

⁽⁹⁾

(u, v) 7→ [˜ u, v] = ˜ ω

^{x∗(y)−y∗(x)}

.

⁽¹⁰⁾

A level3 theta struture for

(A, L)

^{is an}isomorphism

α : H

g

(3) → G(L

^{∼ 3}

)

whihis the identity one restrited to

C

^∗.

Projeting on

( Z /3 Z )

^{2g,a level 3 thetastruture}

α

^induesan isomorphism

˜

α : ( Z /3 Z )

^2g

→

^∼

A[3]

whih is sympletiw.r.t. the Weil pairing on

A[3]

^{and the standard sympleti pairingon}

( Z /3 Z )

^2g

× ( Z /3 Z )

^{2g. Suhan} isomorphismis alled alevel3 struture on

(A, L)

^.

Let

V

3

(g )

^{be the vetor spae of omplex valued funtions over}

( Z /3 Z )

^{g. It is well known,}

by the work of Mumford [Mum66℄, that a level 3theta struture

α

^{indues an}isomorphism (uniqueup toasalar)betweenthe

3

^g-dimensionalvetorspaes

H

⁰

(A, L

³

)

^and

V

3

(g )

^{. This}

allows us to identify

P H

⁰

(A, L

³

)

^{with the abstrat}

P

^3g−1

= P (V

3

(g))

^{and to equip it with}

a anonial basis orresponding to the funtions

{X

_α

} ∈ F unct(( Z /3 Z )

^g

, C )

^{, dened in the}

followingway

X

α

: ( Z /3 Z )

^g

−→ C ,

⁽¹¹⁾

X

α

(α) = 1,

X

α

(σ) = 0

^if

σ 6= α.

There exists only one irreduible representation of

H

_g

(3)

^on

V

3

(g )

^where

C

^∗ ats linearly (this is usually alled a level 1 representation): the so-alledShrödinger representation

U

^.

Let

(t, x, x

^∗

)

^{be anelementof}

H

_g

(3)

^and

X

α

∈ V

3

(g)

^,then

U (t, x, x

^∗

) · X

α

= tx

^∗

(α + x)X

α+x

.

Remark 2.0.2 Let

A

g

(3)

^{be the moduli spae of ppas with a level 3 struture and}

A

g

(3, 6)

themodulispae ofppaswith alevel3strutureandaneven thetaharateristi. Thegroups

Γ

g

(3)

^and

Γ

g

(3, 6)

^{dened in Setion 1 at properly dis}ontinuously and holomorphially on the Siegel upper half-plane

H

₂ induing the isomorphisms

A

g

(3) ∼ = H

_g

/Γ

g

(3)

^and

A

g

(3, 6) ∼ = H

_g

/Γ

g

(3, 6)

^.

2.1 The ation of

ı

Let

(A, H)

^and

L

^{be as in the preeding paragraph and}

φ : L →

^∼

ı

^∗

L

^{be the normalized}

isomorphism. This isomorphismindues involutions

ı

^#

: H

⁰

(A, L

ⁿ

)→H

⁰

(A, L

ⁿ

)

^{for every}

n

^,

dened in the followingway

ı

^#

(s) = ı

^∗

(φ

ⁿ

(s)).

For our goals,it is useful tohave anintrinsi omputationof the dimensions of

H

⁰

(A, L

ⁿ

)

+

et

H

⁰

(A, L

ⁿ

)

−^{,that wewillmakeby meansof the}Atiyah-Bott-Lefshetzxedpointformula ([GH78℄, p. 421). Weknow that the xed points of

ı

^{are 2-torsion points,thus}

X

2 j=0

(−1)

^j

T r(ı

^#

: H

^j

(A, L)) = X

β∈A[2]

T r(ı : L

β

→ L

β

) det(Id − (di)

_β

) .

Now

(di) = −Id

^so

det(2Id) = 2

^{g. Realling setion 1.1, if the symmetri line bundle}

L

^is

even, we have

X

β∈A[2]

T r(ı : L

β

→ L

β

) = 2

^g−1

(2

^g

+ 1) − 2

^g−1

(2

^g

− 1) = 2

^g

,

otherwise

−2

^{g. F}urthermore,as

L

^{representsaprinipal}polarization,

h

^p

(A, L) = 0

^for

p > 0

^.

Therefore, by denition of

H

⁰

(A, L)

+ ^and

H

⁰

(A, L)

−^,

X

2 j=0

(−1)

^j

T r(ı

^#

: H

^j

(A, L)) = h

⁰

(A, L)

+

− h

⁰

(A, L)

−

.

Developing this formulawe nd that, for aneven linebundle representing the polarization,

h

⁰

(A, L)

+

+ h

⁰

(A, L)

−

= 1 h

⁰

(A, L)

+

− h

⁰

(A, L)

−

= 1,

whihimplies

h

⁰

(L)

+

= 1

^and

h

⁰

(L)

−

= 0

^{. If the linebundle isodd, we have}

h

⁰

(A, L)

+

+ h

⁰

(A, L)

−

= 1

h

⁰

(A, L)

+

− h

⁰

(A, L)

−

= −1,

and the dimensions of the eigenspaes are respetively 0 and 1.

If we are instead onsidering the

n

^{-th power of}

L

^{then the parity of}

n

^{omes into play,}

beause

e

^L_∗ⁿ

(x) = e

^L_∗

(x)

^{n. Therefore, if}

n ≡ 0 mod 2

^{, the parity of the line bundle is not}

importantand we have

h

⁰

(A, L

ⁿ

)

+

+ h

⁰

(A, L

ⁿ

)

−

= n

^g

h

⁰

(A, L

ⁿ

)

₊

− h

⁰

(A, L

ⁿ

)

₋

= 2

^g

.

This implies

h

⁰

(A, L

ⁿ

)

+

= (n

^g

+ 2

^g

)/2

^and

h

⁰

(A, L

ⁿ

)

−

= (n

^g

− 2

^g

)/2

^{. If}

n ≡ 1 mod 2

we need to make dierent alulations depending on the parity of the line bundle. These

alulations, that we omit as they ome from onsiderations very similar to the preeding

ones, are summarized inthe following Proposition(BL here means base lous).

Proposition 2.1.1 Let A be an abelian variety of dimension g, n a positive integer and

L

a symmetri line bundle on A s.t.

h

⁰

(A, L) = 1

^{. The}

2

^{2g 2-torsion points are divided into}

two sets dened in the followingway

S

+

:= {x ∈ A[2] s.t. e

^L_∗

(x) = 1}, S

−

:= {x ∈ A[2] s.t. e

^L_∗

(x) = −1}.

If n isodd then, depending on the parity of L, we have:

L even:

1.

#(S

₊

) = 2

^g−1

(2

^g

+ 1)

^and

#(S

₋

) = 2

^g−1

(2

^g

− 1)

^;

2.

h

⁰

(A, L

ⁿ

)

+

= (n

^g

+ 1)/2

^and

h

⁰

(A, L

ⁿ

)

−

= (n

^g

− 1)/2

^.

L odd:

1.

#(S

−

) = 2

^g−1

(2

^g

+ 1)

^and

#(S

+

) = 2

^g−1

(2

^g

− 1)

^;

2.

h

⁰

(A, L

ⁿ

)

+

= (n

^g

− 1)/2

^and

h

⁰

(A, L

ⁿ

)

−

= (n

^g

+ 1)/2

^.

In both ases BL

(|L

ⁿ

|

₊

) = S

−^{, BL}

(|L

ⁿ

|

₋

) = S

+ ^{and the origin}

0 ∈ S

+^.

If n iseven, then

h

⁰

(A, L

ⁿ

)

+

= (n

^g

+ 2

^g

)/2, h

⁰

(A, L

ⁿ

)

−

= (n

^g

− 2

^g

)/2.

Moreover

|L

ⁿ

|

₊ ^{is base point free and BL}

(|L

ⁿ

|

₋

) = A[2].

Proof: Weremark that for every positiveinteger

n

^,

BL (|L

ⁿ

|

+

) ∪ BL (|L

ⁿ

|

−

) = A[2].

Let

n

^{be odd. Sine we use the} linearization given by the normalized isomorphism, the assertionabout the origin is true by denition. It remains to prove the assertionabout the

base lous. We reall that, if

x ∈ A[2]

^,

e

^L_∗

(x)

^{is the salar}

α

^s.t.

φ(x) : L

_ı(x)

∼ = L

_x

→ L

_x ^is

the multipliation by

α

^{. Thus, given an invariant setion}

ϕ ∈ H

⁰

(A, L

ⁿ

)

+ ^and

y ∈ S

−^{, we}

have

ϕ(y) = (ı

^#

(ϕ))(y) = −ϕ(y),

so

ϕ(y) = 0

^{. This implies that all invariantsetions must vanish at points of}

S

−^{. A similar}

argumentshows that allanti-invariant setionsvanish at points of

S

+^.

If

n

^{is even, then we anwrite}

n = 2k

^{for some}

k ∈ N

. Wereall that the linearsystem

|L

²

|

is base point free and that all setions of

H

⁰

(A, L

²

)

^{are invariant. Then the linear system}

Sym

^k

(H

⁰

(A, L

²

))

^{isalsobasepointfreeandby takingtherestritionof}

Sym

^k

(H

⁰

(A, L

²

))

^to