The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let f:S′⟶S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:S'\longrightarrow S$$\end{document} be a cyclic branched covering of smooth projective surfaces over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document} whose branch locus Δ⊂S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \subset S$$\end{document} is a smooth ample divisor. Pick a very ample complete linear system |H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|{\mathcal {H}}|$$\end{document} on S, such that the polarized surface (S,|H|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(S, |{\mathcal {H}}|)$$\end{document} is not a scroll nor has rational hyperplane sections. For the general member [C]∈|H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[C]\in |{\mathcal {H}}|$$\end{document} consider the μn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{n}$$\end{document}-equivariant isogeny decomposition of the Prym variety Prym(C′/C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Prym}\,}}(C'/C)$$\end{document} of the induced covering f:C′:=f-1(C)⟶C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$\end{document}: Prym(C′/C)∼∏d|n,d≠1Pd(C′/C).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$\end{document}We show that for the very general member [C]∈|H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[C]\in |{\mathcal {H}}|$$\end{document} the isogeny component Pd(C′/C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{d}(C'/C)$$\end{document} is μd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{d}$$\end{document}-simple with Endμd(Pd(C′/C))≅Z[ζd]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$\end{document}. In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map Pd(C′/C)⊂Jac(C′)⟶Alb(S′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$\end{document}.

Then, we can state the main result of this paper, which is the following: Theorem 1.1 Let S be a smooth projective surface over C with an ample line bundle L. Assume ∈ |L ⊗n | is smooth and consider the n-fold cyclic covering f : S −→ S branched along the divisor . Given a very ample complete linear system |H| on S, such that (S, |H|) is not a scroll nor has rational hyperplane sections. Then, for the very general member [C] ∈ |H| we have that Especially, each P d (C /C) is a μ d -simple abelian variety.
If we restrict to the case of double coverings, we note that the involution σ of the covering f acts as − id on P 2 (C /C) = Prym(C /C) and thus, End μ 2 (Prym(C /C)) = End(Prym(C /C)). In particular, (1.1) can be stated as follows:

Corollary 1.2 Let S be a smooth projective surface over C with an ample line bundle L. Assume
∈ |L ⊗2 | is smooth and consider the double covering f : S −→ S branched along the divisor . Given a very ample complete linear system |H| on S, such that (S, |H|) is not a scroll nor has rational hyperplane sections. Then, for the very general member [C] ∈ |H| we have that The condition the line bundle L is ample in (1.1) implies that Alb( f ) : Alb(S ) −→ Alb(S) is an isomorphism cf. page 11 and therefore the map P d (C /C) −→ Alb(S ) is trivial. For the general situation one needs to consider the abelian subvariety Then, the result can be reformulated as follows: Theorem 1.3 Let S be a smooth projective surface over C with a line bundle L. Assume ∈ |L ⊗n | is smooth and consider the n-fold cyclic covering f : S −→ S branched along the divisor . Given a very ample complete linear system |H| on S, such that (S, |H|) is not a scroll nor has rational hyperplane sections. Then, exactly one of the following assertions holds true: In this paper we present a complete proof for the above results, inspired by Ciliberto and Van der Geer's approach in [3]. We note that this method does not capture the étale situation, cf. (3.2), (3.3) and (3.4). In addition, if we rephrase the statement for n > 2 by requiring simplicity instead of μ d -simplicity to the isogeny components, we observe that this method cannot be adopted. Namely, the abelian variety B in (3.4) cannot be chosen in general to be μ d -invariant and for this reason the last combinatorial argument in (3.4) fails. Lastly, a result due to Ortega and Lange, cf. [6] may be used to find counter-example for the case the covering f is étale of degree 7. Notations and Conventions. For n ∈ N, μ n is the constant group scheme over C, which is associated to the abstract group Z/nZ. The symbol ζ n stands for a primitive n-th root of unity. If A is an abelian variety over C, which is endowed with a μ n -action, then End μ n (A) is the ring of μ n -equivariant endomorphisms of A. A very general point of a given variety X is a closed point x ∈ X , that lies in the complement of a countable union of nowhere dense closed subvarieties.

Preliminaries
In this section, we state some well-known results, which are needed later.

Proposition 2.1 Let π : A −→ S be a projective abelian scheme over a Noetherian base S. Then, the endomorphism functor of A over S is representable by an S-scheme End A/S , which is a disjoint union of projective and unramified S-schemes.
Proof This is well-known, cf. [4, pp. 133].
The following proposition relates the correspondences on C × C with the endomorphisms of the Jacobian Jac(C).

Proposition 2.2
Let π : X −→ S be a projective smooth morphism over a Noetherian base S, whose fibres are geometrically integral curves. Furthermore, assume that the morphism π admits a section, i.e. X (S) = ∅. Then, there is a natural and functorial isomorphism Corr S (X ):= Pic(X × S X )/(pr 1 ) * Pic(X ) ⊗ (pr 2 ) * Pic(X ) ∼ = End S (Pic 0 X /S ). Proof Consider the commutative diagram: 0 P i c (X )/π * Pic(S) Pic(X × S X )/(pr 2 ) * Pic(X ) Corr S (X ) 0 The first row is clearly exact: Indeed, the relative Picard functor is an fppf-sheaf, cf. [13, Tag 021L], [5,Thm. 2.5] and thus, the restriction map (pr 1 ) * is injective. Furthermore, the map q is just the cokernel of (pr 1 ) * . Next, we give the definition of the map d. Fix x ∈ X (S) and let φ : X −→ Pic X /S be any S-morphism. Then, dφ is the unique endomorphism of Pic 0 X /S , making the diagram below commutative.
Note that under our assumptions the Albanese map can : X −→ Alb X /S exists and has the desired universal property, cf. [ The following proposition is well-known.

Proposition 2.3 Suppose that the polarized surface (S, |H|)
is not a scroll nor has rational hyperplane sections. Then, the following assertions hold true: (i) The discriminant divisor D is irreducible and has codimension one in |H|, i.e. D is a prime divisor of |H|. (ii) The general curve [C] ∈ D is irreducible and has a single ordinary double point as its only singularity.
We close this section by introducing the μ n -equivariant isogeny decomposition in (1.1). Let f : C −→ C be a cyclic branched covering of smooth complex projective curves with deg( f ) = n and let σ stand for a generator of the Galois group of f . The μ n -action on C induces an action on Jac(C ) and thus, it defines a Q-algebra homomorphism For any divisor d|n, we Then, the addition map μ : is a μ n -equivariant isogeny. Lange and Recillas [7] have stated and proved the relation between Q-representations and the G-equivariant isogeny decomposition of an abelian variety with G-action, in terms of the finite group G involved, cf. [7, Thm. 2.2]. The μ nequivariant isogeny decomposition of Jac(C ) given above is in fact identical with the one introduced by Lange and Recillas [7]. This can be seen for example by using [

Reduction to the generic fibre
Let S be a smooth projective surface over C with an ample line bundle L. Assume ∈ |L ⊗n | is smooth and consider the n-fold cyclic covering f : S −→ S branched along the divisor . Furthermore, fix a very ample complete linear system |H| on S, such that the polarized surface (S, |H|) is not a scroll nor has rational hyperplane sections. In this section we reduce the proof of Theorem 1.1 to showing that Let x ∈ S be a closed point of S. We denote by |H| x the linear system of hyperplane sections in |H| passing through x. In the following we impose restrictions on the point x, i.e.
x ∈ S will be taken from some appropriate non-empty open subset of S.
Let g : X ⊂ S × |H| x −→ |H| x denote the universal family of hyperplane sections and h : Y ⊂ S × |H| x −→ |H| x its pullback to S , i.e. Y:=X × S S . Note that over the non-empty open subset U ⊂ |H| x of smooth curves which intersect the branch locus transversally both g and h are smooth families of curves having a section. The latter allows us to consider their families of Jacobians over U , which we denote by p : A generator σ : S −→ S of the Galois group of the covering f induces an automorphism of Y over U and thus, an automorphism σ * : As a first step we use the representability of the endomorphism functor of abelian schemes cf. (2.1) to reduce the proof of Theorem 1.
wherē η is a fixed geometric generic point of |H| x . The proof of this is standard and so we omit it.
Let [C] ∈ |H| x be an irreducible member with a single ordinary double point as its only singularity and intersecting the branch locus transversally. Then, C := f −1 (C) is irreducible and has n ordinary double points as its only singularities. In this case the group variety P d (C /C) is semi-abelian. In particular, the result is the following:

Lemma 3.2 For an irreducible member [C]
∈ |H| x with a single ordinary double point as its only singularity and intersecting the branch locus transversally, there is an exact sequence: where ν :C −→ C is the normalisation map and ϕ(d) is the Euler's totient function.
Proof We have a commutative diagramC wheref is the cyclic covering branched along the divisor ν * | C ∈ |ν * L| ⊗n C | and ν is the normalisation of C . Fix a generator σ of Aut(C /C) and letσ be the corresponding generator of Aut(C /C), i.e. the one for which the diagram below commutes Let {y, σ (y), σ 2 (y), . . . , σ n−1 (y)} be the set of ordinary double points of C . Then, we find a commutative diagram with exact rows and columns We show that β induces a surjection P d (C /C) = ker 0 ( d (σ * )) → → P d (C /C) = ker 0 ( d (σ * )). Indeed, by Snake lemma we have the exact sequence Note that coker(γ ) is an affine algebraic group, as it is the quotient of a commutative affine algebraic group by an algebraic subgroup. Since ker( d (σ * )) is a projective variety and the last arrow in the above sequence is surjective, [14,Cor. 12.67] shows that coker(γ ) is finite. The latter provides the surjectivity of the map ker 0 ( d (σ * )) −→ P d (C /C) = ker 0 ( d (σ * )), as claimed.
We are now in the position to prove the following: Proof The one direction is clear: , then every non-zero μ d -equivariant endomorphism of (P d )η is an isogeny and thus, (P d )η is a μ d -simple abelian variety. Conversely, assume that (P d )η is μ d -simple. We divide the proof into steps.
Step 1. There is a closed subscheme End  Step 2. The fibre (P d ) [C] for the very general member [C] ∈ |H| x is a μ d -absolutely simple abelian variety.

Proof of Step 2
Recall that the U -scheme End μ d P d /U (0) is unramified cf. (2.1). It follows that a geometric fibre of this U -scheme is a disjoint union of points, corresponding to the μ dequivariant endomorphisms of P d , which are not isogenies cf. Step 1. Since (P d )η is a μ d -simple abelian variety, the only μ d -equivariant endomorphism of (P d )η, that is not an isogeny is the zero-morphism. In particular, this means that the geometric generic fibre of the U -scheme End μ d P d /U (0) is connected and therefore, we can determine countably many non-empty open subsets U i ⊂ U , such that the U -scheme End μ d P d /U (0) has (geometrically) connected fibres for all points lying in the intersection of the U i 's, cf. [13, Tag 055C]. Thus, for the very general member [C] ∈ |H| x , the only μ d -equivariant endomorphism of (P d ) [C] , which is not an isogeny is the zero-morphism. The latter is equivalent to the μ d -simplicity of (P d ) [C] , proving the claim.
Pick a Lefschetz pencil (C t ) t∈P 1 ⊂ |H| x . We may assume that all its singular members are irreducible and intersect the branch locus transversally, cf. (2.3).
Step 3. Given a Lefschetz pencil (C t ) t∈P 1 as above, we construct a homomorphism: whereμ is a fixed geometric generic point of P 1 .

Proof of Step 3
Since the endomorphism ring of any abelian variety is finitely generated, cf.
[ [9], Thm. 12.5], we find a finite field extension L ⊃ κ(μ), such that every endomorphism of P d over κ(μ) is defined over L, i.e. End((P d )μ) = End((P d ) L ). Consider the smooth projective model E of L together with the morphism E −→ P 1 induced by this field extension and fix a closed point y ∈ E lying over a point of the pencil that corresponds to a nodal curve. The map ρ : . Then, f extends to an endomorphism over the local ring R of E at the point y, cf. [12,Prop. 7.4.3]. The restriction of the first projection of P d × R P d to the graph of f is an isomorphism. We set α:= pr 1 | ( f ) y . By pulling back α along G To see this, observe that the composite is the zero map by [ [9], Cor. 3.9] and hence, pr 2 | G ϕ(d) m factors through the kernel of (P d ) y −→ m . One checks that ρ is a homomorphism of rings. for n ∈ , we get finitely many nowhere dense closed subsets of U , such that for every point u ∈ U \ n∈ Z n the fibre π −1 (u) contains a point, which is not in Z[ζ d ]. We can choose a Lefschetz pencil as above, such that (P d )μ is μ d -simple, cf.
Step 2 and End μ d ((P d )μ) = Z[ζ d ]. By Step 3 this leads to a contradiction. Indeed, using that every non-zero element of it is readily checked that the composition of the map ρ constructed in Step 3 with ψ:= pr 1 •− : The proof is complete.
The next lemma consists of the final reduction step.

Lemma 3.4 The abelian variety (P d ) η is μ d -simple if and only if it is μ d -absolutely simple.
Proof Clearly, if (P d ) η is μ d -absolutely simple, then it is μ d -simple. Conversely, assume that (P d ) η is μ d -simple but not μ d -absolutely simple. Then, there is a finite field extension L ⊃ κ(η) and a non-zero and proper μ d -simple abelian subvariety B of (P d ) L , such that (P d ) L can be written up to isogeny as a product B τ , where B τ stands for a Galois conjugate of B and τ runs through a finite subset J ⊂ Gal(L/κ(η)) of cardinality greater equal to 2. The field extension L ⊃ κ(η) gives rise to a morphism g : U −→ U , which we may assume is étale. For τ ∈ J , we let ϕ τ be the endomorphism of (P d ) L whose image is B τ . More explicitly, ϕ τ is given by Pick a Lefschetz pencil (C t ) t∈P 1 , such that its singular members are irreducible and intersect the branch locus transversally. Let X be any irreducible component of g −1 (P 1 ∩ U ). Then, X dominates P 1 ∩ U and if θ ∈ X is its generic point, then each ϕ τ determines an endomorphism of P d over θ , e.g. using the Néron mapping property, such that if B τ := im(ϕ τ ), then B τ ∼ (P d ) θ . LetX be a smooth compactification of X andX −→ P 1 the extension of g : X −→ P 1 ∩ U . Fix a point y ∈X lying over a point of the pencil which corresponds to a nodal curve and consider the local ring R ofX at y. Since P d admits a semi-abelian reduction over R , cf. (3.2) the same is true for all B τ , cf. [12,Cor. 7.1.6]. We denote bỹ B τ the identity component of the Néron model of B τ . Then, the isogeny of the generic fibre extends to an isogeny B τ ∼ P d over R, cf. [12,Prop. 7.3.6]. Since (P d ) y is an extension of an abelian variety by a torus of rank ϕ(d), cf. (3.2), it follows that the toric part ofB τ y has rank δ, 1 ≤ δ ≤ ϕ(d), such that δ|J | = ϕ(d). As in Step 3, one constructs a homomorphism ρ τ : has rank ϕ(d) as a free abelian group, we conclude that δ = ϕ(d). But then |J | = 1, which is absurd.

The Proof of Theorem 1.1
According to the results of Sect. 3, our task to prove Theorem 1.1 is reduced to showing (P d ) η is a μ d -simple abelian variety. Recall, that we have an isogeny Given a non-zero endomorphism ε ∈ End μ d ((P d ) η ). Then, by considering the composite ε : Jac(C η ) we get an endomorphism of Jac(C η ) whose restriction to (P d ) η is simply ε • [n]. Hence, it suffices to show that that the restriction of ε to (P d ) η lies in Z[ζ d ]. Recall, that abelian schemes satisfy a stronger Néron mapping property, cf. [10,Sec. 3.1.5]. Thus, the endomorphism ε extends to an endomorphism Let [C] ∈ |H| x be a general member and choose a general two-dimensional linear system containing [C]. Consider the rational map φ ,T . Then, for a general point y ∈ C we get a divisor y = φ ,T (y) on S . Set w = f (y) ∈ C, f −1 (w) = {y, σ (y), . . . , σ n−1 (y)} and f −1 (x) = {z, σ (z), . . . , σ n−1 (z)}, where σ is a generator of the Galois group of the covering f . The following lemma computes the divisor E y in C corresponding to the intersection of C with y . ⊂ S and observe that the map is invariant under the μ n -action. Therefore, we find a homomorphism Alb(S) −→ Alb(S ) that is inverse to Alb( f ), proving the claim. In particular, we deduce that q(S) = q(S ). Here, we give the proof for the case S is regular, i.e. q(S) = 0.

Irregular case
The closed embedding i : C → S defines the natural map i * : Pic 0 (S ) −→ Pic 0 (C ) whose kernel is finite, since H 1 (S , O S (−C )) = 0. In what follows we view Pic 0 (S ) as an abelian subvariety of Jac(C ) by identifying it with im(i * ). We shall use the following lemma. Lemma 4.2 Let a : Jac(C ) −→ P d (C /C) ⊂ Jac(C ) be a homomorphism and let T a be a correspondence associated to it, cf. (2.2). Assume that there exist α 0 , . . . , α n−1 ∈ Z, such that for general y ∈ C the divisor class T a (y −σ (y))+α 0 (y −σ (y))+. . .+α n−1 σ n−1 (y −σ (y)) lies in Pic 0 (S ). Then, the restriction of a to P d (C /C) lies in Z[ζ d ] ⊂ End(P d (C /C)).