AbstractLet $$f:S'\longrightarrow S$$
f
:
S
′
⟶
S
be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$
C
whose branch locus $$\Delta \subset S$$
Δ
⊂
S
is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$
|
H
|
on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$
(
S
,
|
H
|
)
is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$
[
C
]
∈
|
H
|
consider the $$\mu _{n}$$
μ
n
-equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$
Prym
(
C
′
/
C
)
of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$
f
:
C
′
:
=
f
-
1
(
C
)
⟶
C
: $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$
Prym
(
C
′
/
C
)
∼
∏
d
|
n
,
d
≠
1
P
d
(
C
′
/
C
)
.
We show that for the very general member $$[C]\in |{\mathcal {H}}|$$
[
C
]
∈
|
H
|
the isogeny component $${\mathcal {P}}_{d}(C'/C)$$
P
d
(
C
′
/
C
)
is $$\mu _{d}$$
μ
d
-simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$
End
μ
d
(
P
d
(
C
′
/
C
)
)
≅
Z
[
ζ
d
]
. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$
P
d
(
C
′
/
C
)
⊂
Jac
(
C
′
)
⟶
Alb
(
S
′
)
.
The generic isogeny decomposition of the Prym Variety of a cyclic branched covering
