As with proposition 3.20, the proof shows another important fact:
Addendum 3.47. Any indecomposable bundleEonΣαhasπα∗(E)∼=Gr(πα∗E).
In particular, for any two indecomposable bundlesE, E′ onΣα, πα∗(E)is S- equivalent toπα∗(E′)if and only ifπα∗(E)∼=πα∗(E′).
Proof. One may repeat the proof of addendum 3.21, word by word, with Lre- placed byE, andnreplaced bym.
Corollary 3.48. IfEis the graded representative of a point inM(mn, d), thenπα∗(E) is isomorphic to its graded bundle. In particular, ifEandE′are graded representatives of points inM(mn, d)andπα∗(E)is S-equivalent toπα∗(E′), thenπα∗(E)∼=πα∗(E′).
Proof. SupposeE =Ls
i=1Ei, whereEiare stable with slopeµ(E). This means thatπα∗(E) ∼= Ls
i=1πα∗(Ei) ∼= Ls
i=1Gr(πα∗(Ei)), which is isomorphic to its graded bundle.
We may define:
Definition 3.49. ForE∈Mα(mn, d), letϑα(E) = det(πα∗(E)).
We then have the following generalisation of proposition 3.23, at the same time correcting a minor imprecision in lemma 3.4 of [7].3
Proposition 3.50.
ϑα(E) =
Nmα(det(E)) , m odd
Nmα(det(E))⊗L⊗n/2α , m even (3.15) Proof. Again, the proof is more or less the same as the one for 3.23, only substi- tuting the calculation:
det M
ζ′∈µm
ζ′∗(E)
∼= O
ζ′∈µm
ζ′∗(det(E))∼= O
ζ′∈µm
ζ′∗(det(E))
⊗ OΣα.
Identifying the fibres ofL
ζ′∈µmζ′∗(E)withCmn[µm] =C[µm]⊕mn, notice that the action ofµmis simply mn copies of the regular representation, together with the moving of fibres. Hence, the action on the left hand side multiplies with the
n
m’th power of the determinant of the regular representation.
3Whether or not there is a mistake in [7] depends on how ”regular representation” is interpreted.
It might seem that the authors forget the powermn arising in the present proof.
And so the action on the right hand side that makes the isomorphism equi- variant, is the natural one onN
ζ′∈µmζ′∗(detE)tensored with the action onOΣα
induced by(χreg)mn.
Consequently,det(πα∗(E))is given by the descent ofN
ζ′∈µmζ′∗(detE)(with the natural action) tensored with the descent ofΣα×C(with the action induced byχregmn ). The former yieldsNmα(detE). The latter yieldsL⊗χregmn. The proposition follows by lemma 3.18.
Remark3.51.Notice that whenevermdividesn2,L⊗n/2α is trivial, and hence some of the precautions taken in the following are in fact completely vacuous. How- ever, for the sake of simplicity, I will not distinguish this situation from the gen- eral one.
Proposition 3.46 shows that:|M(n,∆d)|α=πα∗(ϑ−1α (∆d)). Introducing propo- sition 3.50, together with the description of the kernel ofNmα in proposition 3.31, we have, choosinga∈ α2 whenmis even, the following disjoint union:
ϑ−1α (∆d) =
det−1 [
ζ∈µm
∆αd⊗Pαζ
, m odd det−1 [
ζ∈µm
∆αd⊗π∗α(La)⊗mn ⊗Pαζ
, m even (3.16)
wheredetdenotes the determinant map:Mα(mn, d)→Picd(Σα).
The following general result implies that the above is actually a composition into connected components.
Lemma 3.52. Let det : Mα(mn, d) → Picd(Σα)be the determinant map. For any path-connected subsetU ⊆Picd(Σα), the inverse image,det−1(U)is path-connected.
Proof. The key element in the proof is the fact that the map: J(Σα) → J(Σα) given byL 7→ L⊗k is a branched covering for anyk ∈ N. This implies that any curveγ : [0,1] → J(Σα)can be lifted to a curveγ˜ : [0,1] → J(Σα)with
˜
γ(t)⊗k=γ(t).
First we show that the fibresdet−1(∆)are connected. GivenE0andE1 in det−1(∆), choose a curve γ : [0,1] → Mα(mn, d) with γ(ν) = Eν, ν = 0,1.
Consider the curveγ¯: [0,1]→J(Σα)given byγ(t) = det (γ(t))¯ −1⊗∆, and letγ˜ be a lift ofγ¯as described above, withk= mn. Nowδ: [0,1]→Mα(mn, d)defined byδ(t) =γ(t)⊗γ(t)˜ is a curve indet−1(∆), connectingE0andE1.
Finally, givenE0 and E1 indet−1(U), choose a curveγ : [0,1] → U with γ(ν) = det (Eν). Again, let ˜γ : [0,1] → J(Σα)be such thatγ(t)˜ ⊗mn = γ(t)⊗ γ(0)−1. Then,δ(t) =E0⊗˜γ(t)is a curve indet−1(U), connectingE0to a point with the same determinant asE1. So, by the above, we connect this point toE1
with another curve running insidedet−1(det(E1)).
Definition 3.53. Whennis odd, for eachζ∈µm, denote:
Bαζ = det−1 ∆αd ⊗Pαζ . Whenmis even, for eachζ∈µmand eacha∈ α2, denote:
Baζ = det−1 ∆αd ⊗π∗α(La)⊗mn ⊗Pαζ .
Remark 3.54. Notice the sudden change of division into cases. In view of the previous, it would seem more obvious to distinguish between whether or notm is odd. However, ifnis odd, so ism. And ifnis even, andmhappens to be odd, mdividesn2, and we have:πα∗(La)mn ∼=OΣα.
So, in all cases we see by the previous results that whennis odd:
πα∗( [
ζ∈µm
Bζα)
and whennis even anda∈ α2:
πα∗( [
ζ∈µm
Baζ)
Remark3.55. The only obstacle left towards identifying the components of the fixed point variety lies in the fact thatµmno longer acts transitively on the fibres ofπα∗. For instance, ifn = 4andm= 2, we may have line bundlesL1andL2
onΣαwith equal degreesd∈ {0,1},πα∗(L1)andπα∗(L2)stable, butζ∗L1being non-isomorphic toL2for anyζ ∈ µr. Thenπα∗(L1⊕L2)∼=πα∗(ζ∗L1⊕L2)∈
|M(4,2d)|α.
This phenomenon sometimes causes the images of theBζa to intersect, even if one is not the pull-back of the other. In the particular case mentioned above, det(ζ∗L1⊕L2) = det(L1⊕L2)⊗ζ∗L1⊗L−11 . Ifd = 0,ζ∗L1⊗L−11 lies inPα1, whenceζ∗L1⊕L2lies in the same component ofdet−1(Nm−1α (OΣα))asL1⊕L2. Hence πα∗(L1 ⊕L2) does not give rise to an intersection between otherwise disjoint components. On the other hand, ifd= 1,ζ∗L1⊗L−11 lies inPαζ, causing the images of all theBζato intersect inπα∗(L1⊕L2).
The above remark actually concludes the proof of theorem 3.44 in the case n= 4upon observing that the casem= 1is trivial. For general values ofn, one needs to consider all possible gradings of a semistable bundle inϑ−1α (∆d).
Proposition 3.56. Letd˜= (d,dn
m). Let˜r= (m,d)˜ andq˜= mr˜. Whennis odd, forζ1, ζ2∈µm:
πα∗(Bζα1)∩πα∗(Bαζ2)6=∅ ⇔ζ1
ζ2
∈µ˜q
Whennis even, fora∈ α2 andζ1, ζ2∈µm:
πα∗(Bζa1)∩πα∗(Baζ2)6=∅ ⇔ζ1
ζ2
∈µ˜q
Proof. The proof is exactly the same in both cases, except for the difference in notation. We will do it fornodd.
Suppose first thatE1∈Bζα1,E2∈Bαζ2andπα∗(E1)is S-equivalent toπα∗(E2).
We may assume without loss of generality thatE1andE2are isomorphic to their graded bundles, and hence by corollary 3.48,πα∗(E1)∼=πα∗(E2).
SupposeEi =Lsi
j=1Ei,j, whereEi,jis stable and has the same slope asEi. Sinceπα∗(E1)∼=πα∗(E2), we get by pulling back toΣαthat:
M
ζ∈µm
s1
M
j=1
ζ∗(E1,j)∼= M
ζ∈µm
s2
M
j=1
ζ∗(E2,j)
Consequently, by the theorem of Krull-Remak-Schmidt, each of theE1,j is isomorphic to a pull back of one of theE2,j. We may assume (handling the general situation recursively) thatE1,1 ∼= ζ∗(E2,1)andE1,j ∼= E2,j forj > 1.
Now,
det(E1) = Os j=1
det(E1,j)∼=ζ∗(det(E2,1))⊗det(E2,1)−1⊗det(E2)
Supposedeg(E2,1) =d1,rk(E2,1) =r1. SinceE2,1has the same slope asE2, we have:
d1=r1d
n m
=r1
n m
(d,mn) −1
d (d,mn) which can be any integer multiple ofd˜= (d,dn
m).
By proposition 3.31,ζ∗(det(E2,1))⊗det(E2,1)−1 ∈ Pαζd1, and hence: ζ1 = ζd1ζ2.It remains only to realize thatζd1∈(µm)d˜=µ˜q.
Conversely, using the considerations above, givenζ1, ζ2 ∈µmwith ζζ12 ∈µq˜, it is straightforward to construct one’s own bundlesEi ∈Bζαi withπα∗(E1) ∼= πα∗(E2), by applying the considerations above.
Definition 3.57. Letζ ∈ µ˜r. Whennis odd, denote by|M(n,∆d)|ζα the com- ponent of|M(n,∆d)|α that containsπα∗(Bαζ′)for any (and hence all)ζ′ ∈ µm
withζ′q˜=ζ. Similarly, whennis even, denote by|M(n,∆d)|ζathe component of
|M(n,∆d)|αthat containsπα∗(Baζ′)for any (and hence all)ζ′ ∈µmwithζ′q˜=ζ.
To prove theorem 3.44, it remains only to examine how the definitions de- pend on the choice ofa, whenmis even.
Proposition 3.58. Assume thatnis even and leta1, a2 ∈ α2. Lets= mn,˜r= (m,d)˜ andq˜=m˜r, whered˜= (d,dn
m). Furthermore, letλ=λ2m(a1, a2)∈µ2. Then for eachζ∈µr˜:
|M(n,∆d)|ζa1 =|M(n,∆d)|λa2s˜qζ.
Proof. By proposition 3.32, πα∗(L⊗sa1−a2) ∈ Pαλs, where λ = λm(α, a1−a2) = λ2m(a1, a2). This shows that forζ′∈µm:Bζa′1 =Baλ2sζ′.
Finally, most applications, including the way in which the remaining ele- ments ofJ(n)permute the components, become clear from the following obser- vation.
Proposition 3.59. Letα∈J(n)be primitive andk∈ {0,1, . . . , n−1}. Letr= (n, d), q= nr. Letm= ord(kα), andd,˜˜r,q˜as before. Finally, let˜k= ˜qkq .
Ifnis odd, we have for allζ∈µr:
|M(n,∆d)|ζα⊆ |M(n,∆d)|ζkαk˜. Ifnis even, we have for allζ∈µrand alla∈ α2:
|M(n,∆d)|ζa⊆ |M(n,∆d)|ζ
k˜
ka.
Proof. First of all, we may assume thatkdividesn. (Indeed, in the general case, k= (n,k)k (n, k), and (n,k)k being prime ton, corollary 3.43 reduces the statement of the proposition to the one withαreplaced by(n,k)k αandkreplaced by(n, k).)
Letα˜=kα, which is of order nk. Consider the diagram of coverings:
Σα π //
πα
A
AA AA AA
A Σα˜
πα˜
~~}}}}}}}}
Σ
(3.17)
whereπis thek-sheeted covering associated toπ∗α˜(Lα). Recall from the discus- sion in the proof of proposition 3.20 that the action ofζ∈ µnonΣαcovers the action ofζk onΣα˜.This implies thatNm(Pαζ) =Pα˜ζk whereNmis the norm map associated toπ. Furthermore, we may arrange thatNm(∆αd)∼= ∆αd˜.
Thus, in the case wherenis odd, givenζ∈µn, for anyL∈∆αd⊗Pαζwe have:
πα∗(L)∈ |M(n,∆d)|ζαq
whereq= (n,d)n . But at the same time,πα∗(L)∼=πα∗˜ (π∗(L)),where det(π∗(L))∼= Nm(L)∈∆αd˜⊗Pα˜ζk.
This shows thatπα∗(L)∈ |M(n,∆d)|ζkαqk˜ =|M(n,∆d)|(ζq)
˜k
kα .
In the case wherenis even, anda∈ α2, we may do almost the same: Given ζ∈µn, for anyL∈πα∗(La)⊗∆αd ⊗Pαζ we have:
πα∗(L)∈ |M(n,∆d)|ζaq
whereq= (n,d)n . But at the same time,πα∗(L)∼=πα∗˜ (π∗(L)), where det(π∗(L))∼= Nm(L)∈πα∗˜(La)k⊗∆αd˜⊗Pα˜ζk, ifkis odd, and
det(π∗(L))∼= Nm(L)⊗πα∗˜(Lα)k2 ∈πα∗˜(La)2k⊗∆αd˜⊗Pα˜ζk, ifkis even.
Sinceπα∗˜(La)k∼=π∗α˜(Lka)k, whenkis odd, andπα∗˜(La)2k ∼=OΣα˜ ∼=π∗α˜(Lka)k, whenkis even, we get in both cases that
πα∗(L)∈ |M(n,∆d)|ζka˜qk=|M(n,∆d)|(ζq)
˜k
ka .
Corollary 3.60. Letα, δ˜ ∈J(n), such thatord(˜α) =m. The action ofδonM(n,∆d) induces a permutation on the set of connected components in |M(n,∆d)|α˜. Denote
˜
r= (m, d)andq˜=m˜r. Whennis odd, we have forζ∈µ˜r: Lδ⊗ |M(n,∆d)|ζα˜=|M(n,∆d)|λα˜n( ˜α,δ)q˜ζ Whennis even, we have forζ∈µranda∈ α˜2:
Lδ⊗ |M(n,∆d)|ζa =|M(n,∆d)|λan( ˜α,δ)q˜ζ Proof. Chooseα∈J(n)primitive, such thatα˜=kαfork=mn.
Now, forE ∈ |M(n,∆d)|ζα ⊆ |M(n,∆d)|ζ
k˜
kα, we have by proposition 3.42 thatLδ⊗Elies in|M(n,∆d)|λαqζ ⊆ |M(n,∆d)|(λqζ)
k˜
kα =|M(n,∆d)|λk˜qζ
k˜
kα , where λ=λn(α, δ), and henceλk =λn(˜α, δ).
Finally, as an apology for the notation, I supply an overview of the values of the different variables in the some particular cases. In each case, assuming
˜
α=kα, whereα∈J(n)is primitive, and applying the results of the section to
˜
α, which is of orderm= (n,k)n .
Remark3.61. We have in the following cases:
m=n: d˜=d,r˜=r,q˜=q, andk˜=k.
d= 0: d˜= 0,˜r=m,q˜= 1 =q, and˜k=k.
(n, d) = 1: d˜=d,r˜= 1 =r,q˜=m, and˜k= mkn . n= 4,d= 2,m= 2: d˜= 1,˜r= 1,q˜= 2,˜k=k.
n= 6,d= 4,m= 3, (2|k): d˜= 2,˜r= 1,q˜= 3,˜k=k.
n= 6,d= 4,m= 2, (3|k): d˜= 4,˜r= 2,q˜= 1,˜k= k3.
Intersections of fixed point varieties
Given different elements inJ(n)(Σ), how do their fixed point varieties inter- sect? In the cases where the fixed point varieties are not connected, how do the individual components intersect? These questions are addressed below. We immediately restrict our attention to primitive torsion elements. The results generalise the ones of section 6 in [1].