General birationality and Hyperelliptic Theta divisors

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of a result obtained jointly with Luca Cesarano: that, for a general pair $(A,X)$ of an (ample) Hypersurface $X$ in an Abelian Variety $A$, the canonical map $\Phi_X$ of $X$ is birational onto its image if the polarization given by $X$ is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions.


Introduction
One of the main general problems in algebraic geometry is the study of the canonical and pluricanonical maps of varieties of general type, especially the problem of establishing their birationality, see for instance [Bom72], [Cil97], [Chen98], [CCCJ21], just to name a few items.
We describe here a simple and relatively general method for establishing birationality of a rational map for the general variety in a family, via specializations, see Theorems 2.4 and 2.7.
The applications can be many (see [Cat99] in the case of surfaces), but we focus here on the use of this method for the problem which was our original motivation, and we exhibit a new self-contained proof (Theorem 6.2) of the main result of .The present proof couples this method with an interesting study, of the geometry of Hyperelliptic Jacobians and of some of their unramified cyclic coverings.
In the course of doing this we establish some general results on the graded rings of Theta Functions on Hyperelliptic Jacobians, see Theorems 4.3 and 4.6.

Openness of birationality and general birationality
As already mentioned, our present main problem is: given a family of varieties {X t } t∈T , and a family of morphisms f t : X t → Y t (respectively, rational maps), when can we conclude, from the fact that f 0 is birational onto its image Y 0 , that, for general t, f t is birational onto Y t ?
Let us start with a negative example: let X be a hypersurface in P N of degree d, let P be a point, P ∈ P N , and consider the projection with centre P , π P : P N \ P → P N −1 .
If the hypersurface X 0 has multiplicity d − 1 at the point P , then π P induces a birational map between X 0 and P N −1 , but for a general X the projection is not birational, having degree equal to d − mult P (X), which is ≥ 2 as soon as mult P (X) < d − 1.
The important feature of this example, which motivates the assumption in the following theorems, is that X 0 is a uniruled variety, indeed it is a rational variety: and this must be avoided.
The next example, instead, clarifies the hypotheses needed for the validity of an assertion made in the first version of this paper (see for instance the next Proposition 2.2).
Hence X 0 , X are normal (being hypersurfaces in a smooth manifold).Blowing up the only singular point, we get Z → X ⊂ P N × P 1 , p : Z → P 1 , and the fibre Z 0 consists of the union of the blow up X ′ 0 of X 0 in P , together with the hypersurface Z ′ 0 in the exceptional P N , Z ′ 0 := {φ m (x 1 , . . ., x n ) + λ m g(P ) = 0}, where we assume that P = (1, 0, . . ., 0) and that φ is the leading term of the Taylor development of f at P .For m ≥ N +2 and φ general, Z ′ 0 is a smooth variety with ample canonical system, and X ′ 0 ∩ Z ′ 0 = {φ m (x 1 , . . ., x n ) = 0}, the exceptional divisor of π : X ′ 0 → X 0 .The following Proposition is a direct consequence of Hironaka's II Main Theorem in [Hiro64] Proposition 2.2.Assume that we have a 1-dimensional projective family p : S → T where S is smooth of dimension n + 1, T is a smooth connected curve, 0 ∈ T , and we have a rational map Then there exists a modification π : Z → S such that, setting and denoting by Y the closure of the image of F , (i) Z is smooth, (ii) ) all the fibres of p ′ := p • π : Z → T consist of the union of the strict transform S ′ t of S t := p −1 (t) with other ruled components.(iv) In particular, if the indeterminacy locus of f is contained in S 0 , then T is the normalization of the graph of F , then the fibres Γ t consist of the strict transform of S t plus some uniruled components.
(vi) Shrinking T , we may assume that in (iii) and (v) other uniruled components only occur for t = 0.
Proof.A preliminary observation is that, since dim(S) = n + 1, the indeterminacy locus of F does not contain any fibre S t .
Assertions (i) and (ii) follow from Hironaka's II Main Theorem of [Hiro64] (see especially page 140, and the affirmative answer to Question (F), (iii), the assertion that f r is a morphism, and that the centres D i are smooth and contained in the indeterminacy locus of f i ) ensuring that, via a sequence of blow ups with smooth centres, we get π : Z → S such that the rational map F ′ := F • π becomes a morphism on Z.
For (iii) we just need to observe that, if we blow up a submanifold W of a manifold M , then the blow up M contains as exceptional divisor the ruled manifold P(N W |M ).Hence the exceptional divisors are all ruled, hence so are the new irreducible components of the fibres of p ′ (as they are divisors in Z by our assumptions).
(iv) follows since the centres of the blow up are contained in the inverse image of the indeterminacy locus in S.
(v): since Z t surjects onto Γ t , the other components of Γ t are images of a ruled manifold, hence they are uniruled.
(vi) first of all, the set of t ∈ T such that S t is not irreducible is closed; furthermore, since there is only a finite number of exceptional divisors, there is only a finite number of t such that the fibres Z t and Γ t are not irreducible.So we omit these two finite subsets of T .
Remark 2.3.In view of Hironaka's extension [Hiro77] of the resolution results to complex spaces, one can replace the hypothesis that we have a projective family by the hypothesis that we have a proper family.
Theorem 2.4.(Openness of birationality) Let p : X → T be a 1-dimensional (flat) family of reduced projective subschemes of dimension n (i.e., X is irreducible and T is a smooth connected curve, 0 ∈ T ) such that X 0 = p −1 (0) contains a unique irreducible component X ′′ 0 which is not uniruled.Let f : X P N be a rational map such that f 0 : Assume moreover (**) setting F := f × p : X P N × T , letting Y be the closure of the image of F , and letting Γ be the normalization of the graph of F , then the fibres Γ t are irreducible for t = 0, while Γ 0 consists of the strict transform of X 0 plus some uniruled components.
Then f t : X t → Y t is birational to its image for all t in a neighbourhood of 0 ∈ T .
Proof.Clearly Y is irreducible and it has dimension n + 1 since its fibre Y 0 over 0 contains Y ′ 0 which has dimension n; the fibre Y t over t = 0 contains the image Y ′ t of X t which by assumption is irreducible.The rational map F induces a surjective morphism F ′ : Γ → Y. Γ is irreducible of dimension n + 1, and the central image Y 0 is the image of Γ 0 under a proper map, and contains Y ′ 0 as a component, since the strict transform X ′ 0 of X ′′ 0 is a component of Γ 0 .The other components of Γ 0 are uniruled, hence they cannot dominate the component Y ′ 0 , which is not uniruled.Hence the general point y ∈ Y ′ 0 is in the image of only one point x, this point x lies in X ′ 0 , and the map F ′ is of maximal rank in x, hence a formal isomorphism with its image: because f 0 is a local isomorphism and p is a submersion at x (in particular there is no ramification of F ′ at x).
Consider now a local holomorphic section Σ of Y → T passing through y (which is a smooth point of Y and of the fibre Y 0 , since F ′ and p are local submersions at x).
If the map f t were non birational for all t, then f ′ t : X ′ t → Y ′ t would have positive degree, and would be étale outside of a branch locus B t ⊂ Y ′ t .We have seen that if y ∈ Y ′ 0 is chosen general, it is not contained in the closure B of the branch loci: since there is no ramification at x.
Therefore the inverse image of Σ consists of holomorphic arcs, in a number strictly greater than one, of which only one contains x in its closure, while the other arcs tend to a point z in Γ 0 different from x.
The conclusion is that F ′ (z) = y, z = x, and we have reached the desired contradiction: hence we have proven that f t is birational.
Remark 2.5.The above Theorem and the following ones can be stated in more general situations.
(i) We can consider more generally1 a family X → T of compact complex spaces, and a meromorphic map f : X M , where M is a complex manifold: the above proof works without any change.
(ii) The same theorem is true for a projective family over an algebraically closed field of any characteristic, if we assume that f 0 is separable and birational on X ′′ 0 .We have in fact that F ′ : Γ → Y proper, hence there is a closed set B ⊂ Y with nontrivial complement Y \ B, such that, over Y \ B, F ′ is finite with all the fibres of cardinality d.
If f t is not birational, then d ≥ 2. Since we have shown that y / ∈ B, and that F ′−1 (y) is a single point with multiplicity 1, it follows then that d = 1, a contradiction.
(ii) The theorem was applied as self evident in the case of canonical maps of algebraic surfaces in [Cat99], but its use was criticized as non self evident in [Lied03].All details of the proof are now following from Proposition 2.2, (v), applied to the family of smooth minimal models of surfaces of general type, and from Theorem 2.4.
Before moving to a more general Theorem, we need to state a simple group theoretical result.
Lemma 2.6.Given finite groups Π X < Π ′ < M 0 , where the maximal normal subgroup of M 0 contained in Π X is the identity, let us set: (1) M ν 0 := M 0 /K, where K is the maximal normal subgroup contained in Π ′ , so that (2) M 0 acts faithfully on the coset space F X := M 0 /Π X , whose cardinality will be denoted by d, (3) M ν 0 acts on the coset space F ν := M 0 /Π ′ , whose cardinality will be denoted by m, (4) M t := Π ′ /K t acts on F t := Π ′ /Π X , where K t is the largest normal subgroup of Π ′ contained in Π X .
Then d = δm, where δ is the cardinality of the set And the action of M 0 preserves the blocks corresponding to the m elements of F ν , Hence we have exact sequences and, setting G := Π ′ /K, G < M ν 0 . .With a similar proof to Theorem 2.4, we obtain the following more general result which is useful for applications.
Theorem 2.7.Let p : X → T be a 1-dimensional (flat) family of projective varieties of dimension n, with X irreducible, T a smooth connected curve, 0 ∈ T , such that X 0 = p −1 (0) is irreducible normal.
Let f : X P N be a rational map such that f 0 : X 0 Y ′ 0 is of degree d to its image Y ′ 0 , which is not uniruled.Assume moreover (**) setting F := f × p : X P N × T , letting Y be the closure of the image of F , and letting Γ be the normalization of the graph of F , then the fibres Γ t are irreducible for t = 0, while Γ 0 consists of the strict transform of X 0 plus some uniruled components.
Then (i) f t : X t → Y t has degree δ onto its image for all t in a neighbourhood of 0 ∈ T , with δ dividing d, so that we may write d = mδ.
More precisely, f 0 : X 0 Y ′ 0 admits a factorization as ν 0 • F ′′ 0 , where ν 0 has degree m, and the monodromy group M 0 ⊂ S d of f 0 is thus related to the monodromy group of f t , M t ⊂ S δ , and the monodromy group M ν 0 ⊂ S m of ν 0 , as in the statement of Lemma 2.6.
(ii) In particular, if the monodromy group Proof.Using the same notation as in the proof of Theorem 2.4, we are then in a similar situation.
The general point y ∈ Y 0 is in the image of exactly d smooth points x 1 , . . ., x d of Γ which lie in X ′ 0 , and the map F ′ is of maximal rank in each x i .
What may now change is that y could be contained in the singular locus of Y, and there may be m smooth branches of Y passing through y.
Therefore, we take the normalization ν : Y n → Y, and notice that we have a factorization of F ′ as ν • F ′′ , where F ′′ : Γ → Y n .We observe then that the morphism F ′′ |X ′ 0 will have degree δ onto its image, where d = δm.Hence the d points are grouped in m subsets, corresponding to the inverse images of the points y 1 , . . ., y m lying over y in Y n , and the previous argument using the local holomorphic sections Γ i of Y n → T passing through y i for i = 1, . . ., m shows that the degree of f t : X t → Y t equals δ.
Assertion ii) follows right away because, if m = 1, d, then there is a partition of {1, . . ., d} in m subsets which are permuted by M 0 .Now, the monodromy of f t : X t Y t will be the same as the one of Y 0 is a composition, it follows that the monodromy of f 0 is as claimed, in view of the previous Lemma 2.6, where we divide the respective fundamental groups by the largest normal subgroup of the fundamental group of the open set of X ′ 0 where all coverings are unramified, so that M 0 and M t are the monodromy groups we are talking about.
Corollary 2.8.Let p : X → T be a family of projective varieties of dimension n, where T is smooth and connected.Assume moreover that we are given a rational map f : X P N which is a morphism for t ∈ V , where V is an open set V ⊂ T .
(I) Assume that for a general point t ∈ T there are several 1-parameter specializations, for j = 1, . . ., r, with base T j containing t and t j ∈ T , of the fibre X t = p −1 (t) to the fibre X t j .Assume that these are, as in Theorem 2.7, such that X t j is irreducible and normal with monodromy in S d(t j ) , and that moreover, writing d j := d(t j ), we have Then, for general t, f t is birational.(II) The same conclusion holds if there are two 1-parameter specializations, one such that the monodromy group M 0 ⊂ S d 1 is primitive, the other such that d 1 does not divide d 2 .
Proof.We denote as above by δ the degree of f t for general t.
Our claim(I), in the notation of Theorem 2.7, is that δ = 1, which is obvious since, by (i) of theorem 2.7, d j = δm(t j ) therefore δ divides all the integers d j , hence their GCD.
To show (II), simply apply (i) and (ii) of Theorem 2.7: in fact the general degree δ is either 1 or d 1 by virtue of (ii), while (i) implies that δ|d 2 .Remark 2.9.One can obtain other more complicated criteria using the above exact sequences of groups.
But, if If all specializations found yield a group M t 0 (j) which is Abelian, then a criterion of triviality of M t follows from a criterion similar to the above Corollary, analyzing the primary decompositions of all the groups M t 0 (j) .
If we get one specialization such that one M t 0 is Abelian, then M t is Abelian, and is, for any other specialization, a quotient of the Abelianization of Π ′ by the image of Π X .
Remark 2.10.The main conjecture raised in [Cat-Ces21] is that the canonical map of a general pair (A, X) of an ample hypersurface in an Abelian variety is an embedding if the Pfaffian of the Polarization given by X is at least dim(X) + 2.
Also for this purpose it would be useful to establish in a similar way some criteria guaranteeing 'general embedding', that is, embedding for a general variety in a family.

Theta divisors of Hyperelliptic curves
We begin with a quite elementary result in group theory.
Lemma 3.1.Consider the Group G of the Hypercube, namely the natural semidirect product (induced by coordinates permutation) For instance, σ(e i ) = e σ(i) .
If H ′ is as in (i) and Proof of the claim: it is obvious that the four above subspaces are invariant.
For such an invariant subspace V , assuming that V = 0, consider a vector v of minimal weight w(v) := |{i|v i = 0}|.Denote by w the minimal weight: Otherwise, we may assume, after a basis change, Then e 1 +e 2 , e 2 +e 3 , . . ., e n−1 +e n ∈ V , hence V is an invariant hyperplane.Hence V is orthogonal to a projectively invariant vector, and we see that Passing to (ii), Since the last equality is between an element of H and one of K, this means that σ = τ and v = σ(v); if this is to hold for each v ∈ K, then σ is the identity.
We now come to an important geometrical occurrence of the group G of the Hypercube.
Let C be a Hyperelliptic curve of genus g, and let ψ : C → P 1 be the canonical double cover (such that the canonical map φ of C is the composition φ = v g−1 • ψ, where v g−1 : P 1 → P g−1 is the Veronese embedding of P 1 as a rational normal curve of degree g − 1).
Then, setting Y := C n , the group G acts on Y = C n , and we have the following commutative diagrams: It is well known that for n = g we have Jacobi inversion, that is, C (g) has a surjective birational morphism (the Abel Jacobi map) to the Jacobi variety Jac(C) ∼ = P ic g (C), while for n = g − 1, again via the Abel Jacobi map, C (g−1) has a birational morphism onto the Theta divisor Θ C ⊂ Jac(C).We shall need to make these birational statements more precise.
We have the following classical result, due to Andreotti [Andr58].
Theorem 3.2.If C is a Hyperelliptic curve, then ν : C (g−1) → P g−1 is the composition of the birational Abel-Jacobi map α g−1 : C (g−1) → Θ C with the Gauss map µ C of Θ C , and p : C g−1 → C (g−1) yields the Galois closure of the Gauss map.
For a non hyperelliptic curve C, letting φ be the canonical map φ : C → P g−1 , the composition µ C • α g−1 is the g − 1 secant map of φ(C), and the branch locus of the Gauss map is the dual variety φ(C) ∨ of the canonical curve in (P g−1 ) ∨ .
The monodromy group of the Gauss map equals the monodromy group of the canonical curve φ(C), the symmetric group S 2g−2 .
Proof.As shown by Andreotti, the map π • Ψ is given as follows: where φ is the canonical map φ : C → P g−1 , which is indeed the projective derivative D(φ) of the Albanese map= first Abel Jacobi map α : C → Jac(C).
On the other hand, the Gauss map associates to a point x ∈ Θ C , the Hyperplane spanned by φ(P 1 ), . . ., φ(P g−1 ) since φ is the projective derivative of α.This shows that ν factors as claimed through the Gauss map µ C .The assertion on the Galois closure follows now from Lemma 3.1.See [Andr58] and [ACGH85] page 111 for the last assertions.
The fact that the degree of µ C equals to 2 g−1 follows algebraically since π • Ψ = ν • p, but also geometrically since each hyperplane intersects φ(C), image of P 1 through the Veronese map of degree (g − 1), in exactly (g − 1) points.
For a non hyperelliptic curve the degree equals 2g−2 g−1 .For a general Theta divisor in a principally polarized Abelian variety the degree of the Gauss map is instead equal to g!.
For more general Jacobians, the Gauss map of the Theta divisor is a rational map whose degree was studied in [CGS17].
Proposition 3.3.If C is a Hyperelliptic curve the map ν : C (n) → P := P g−1 has a branch locus B which set theoretically equals the union of ∆, the Discriminant Hypersurface for divisors in P 1 of degree n (the dual variety of the rational normal curve Γ = Γ n ), with 2g + 2 Hyperplanes H 1 , . . ., H 2g+2 , where, if z i is a branch point of C → P 1 , then H i corresponds to the divisors containing z i .Moreover ∆ occurs with multiplicity 2 n−1 in the branch locus, while the divisors H j occur with multiplicity 2 n−2 .
The map ν factors exactly through two intermediate coverings: where ι is the hyperelliptic involution; ii) Z n → Zn , where Zn → P is the double cover branched on the union H of the hyperplanes H 1 , . . ., H 2g+2 .
Proof.In view of Lemma 3.1 the main remaining point to show is that the branch locus is as stated.
The ramification locus of Ψ : C n → (P 1 ) n equals the union of the divisors where R C = {p 1 , . . ., p 2g+2 } is the ramification divisor of ψ : C → P 1 .These divisors are permuted by S n , and their image in (P 1 ) n equals where B C = {z 1 , . . ., z 2g+2 } is the branch divisor of ψ.
Whereas the ramification of π : (P 1 ) n → P n consists of the fixpoints for some nontrivial element of S n , and its image is the discriminant hypersurface ∆ consisting of the nonreduced divisors on P 1 , that is, the divisors x 1 + • • • + x n where the points x i are not distinct.∆ is irreducible, being the image of Hence the branch locus of ν • p = π • Ψ is equal to the union of ∆ and of hyperplanes H 1 , . . ., H 2g+2 , where H i consists of the effective divisors in P 1 containing z i .
H i intersects ∆ in the linear space of codimension 2 consisting of the divisors which are ≥ 2z i , and in a smaller discriminant ∆ ′ i consisting of divisors which are the sum of z i with a nonreduced divisor.
On the other hand, the branch locus of p : C n → C (n) equals the discriminant ∆ C , consisting of nonreduced effective divisors of degree n on C.
∆ C maps then to ∆ with degree 2 n−1 , since for general x 1 and general x 3 , . . ., x n the inverse image of 2x While the inverse image of the ramification of π contains the S n -orbit of the divisors y ′ 1 + y ′′ 1 + y 3 + • • • + y n , where y ′ 1 + y ′′ 1 is the inverse image of x 1 , and y j → x j .
Therefore the branch locus of ν consists of ∆ with multiplicity 2 n−1 , and, since for j ≥ 3 there are two choices for y j , of the hyperplanes H 1 , . . ., H 2g+2 with multiplicity 2 n−2 .
Concerning assertion i), observe that the element e = e i ∈ V acts on C n via the hyperelliptic involution ι acting on each coordinate, hence the intermediate quotient is the quotient of the symmetric product C (n) via the action of ι.
For assertion ii), we notice that the quotient of C n by the subgroup of K orthogonal to e is the double covering of (P 1 ) n branched on the union of the branch divisors B C (i), whose image in P n is the union of the hyperplanes H j .
We can rephrase the previous result in the special case n = g − 1: Proposition 3.4.If C is a Hyperelliptic curve the Gauss map µ C : Θ C → P := P g−1 has a branch locus B which set theoretically equals the union of ∆, the Discriminant Hypersurface for divisors in P 1 of degree g − 1 (the dual variety of the rational normal curve Γ = Γ g−1 ), with 2g +2 Hyperplanes H 1 , . . ., H 2g+2 , where, if z i is a branch point of C → P 1 , then H i corresponds to the divisors containing z i .Moreover ∆ occurs with multiplicity 2 g−2 in the branch locus and the Hyperplanes H j occur with multiplicity 2 g−3 .
The Gauss map µ C factors exactly through two intermediate coverings: where Z → P is the double cover branched on the union of the hyperplanes H i .
Proof.We just need to observe that the hyperelliptic involution ι acts on the Jacobian Jac(C) as multiplication by −1, for a suitable choice of the origin as a thetacharacteristic.
The conclusion is that a hyperplane H is in the branch locus if H intersects Γ in a divisor which is the image of a canonical divisor of C which contains a ramification point p i , or contains a divisor of the form x ′ + x ′′ , the inverse image of a point x ∈ P 1 : this amounts to saying that H intersects Γ in a divisor containing a branch point z i or containing a point x with multiplicity at least 2.
For further purposes, we must clarify the different roles played by the discriminant ∆ and the union of Hyperplanes H 1 ∪ • • • ∪ H 2g+2 in the branch locus.
To quickly get an understanding of this issue, let us consider the case g = n: then the double covering Zg is a variety with trivial canonical divisor, while Z is birational to the Kummer variety of the Jacobian.Hence the map Z → Zg is unramifed in codimension 1.The main point is, as we are now going to explain, that ∆ contributes to an exceptional divisor on the symmetric product of the curve.

Theta functions on Hyperelliptic Jacobians
Let C be a curve of genus g, and let Indeed, every divisor of degree g is effective, and, if we fix a point y 0 ∈ C, we have the Abel Jacobi maps The Abel-Jacobi maps factor through the symmetric products C (n) = C n /S n , and to simplify notation we shall use the same symbol for all of them.We denote also as usual For many assertions we are going to make, see [ACGH85] pages 250 and around it.By Riemann's singularity Theorem, if u 0 = α(D), D ∈ C (g−1) , then there is a thetacharacteristic K such that Up to a translation, we may assume Hence the classical result that is surjective, birational and locally invertible outside in fact for such divisors D ′ there exists D ′′ ∈ |D ′ | with D ′′ ≥ y 0 .
Corollary 4.1.The graded ring of Hyperelliptic Jacobian Theta Functions Hence in this approach it is necessary to study the divisor α −1 (W g−1 ), which contains the divisor y 0 + C (g−1) .Remark 4.2.(1) α −1 (u), for u = α(D), and D an effective divisor of degree g − 1, is the linear system |D|, whose dimension is classically denoted by r.
Since D is a special divisor, it follows by Clifford's Theorem that r ≤ g−1 2 , equality holding if and only if C is hyperelliptic and D is a multiple of the hyperelliptic divisor H.
Its intersection with W g−1 has dimension equal to g − 3 and is contained in the singular locus Sing(W g−1 ).
where the divisor E is exceptional for α.
Proof.Assume that there is a divisor D inside C g which is contracted under the Abel Jacobi map α to a lower dimensional variety.
For convenience, we choose now the base point y 0 ∈ C to be a Weierstrass point, that is, a fixpoint for ι: this means that 2y 0 ∈ |H|, 2y 0 = ψ −1 (x 0 ).Remark 4.5.(a) The divisor Ĉ(g−1) is invariant for the Hypercube group G, actually , where H 0 is the hyperplane in P g of divisors containing x 0 .
(b) We observe here that the divisor E maps onto the discriminant ∆ ⊂ P g−1 under the map ν.
(c) The big diagonal ∆ ′ ⊂ (P 1 ) g , the inverse image of the Discriminant Hypersurface, has the property that Ψ −1 (∆ ′ ) = Ê ∪ ∆ ′ C , where ∆ ′ C is the big diagonal in C g .Ê and ∆ ′ C alone are not G-invariant.Theorem 4.6.The graded ring of Hyperelliptic Jacobian Theta Functions is a subring of invariants as follows: Proof.The first equality is the same equality stated in Corollary 4.1, in view of Theorem 4.3.
For the second equality we need to observe that Ĉ(g−1) + Ê is the pull back of the divisor C(g−1) + E, and that the ramification divisor of C g → C (g) is the big diagonal ∆ ′ C , mapping to the irreducible discriminant divisor ∆ C which is not contained in the divisor C(g−1) + E.
Holomorphic sections downstairs (on C (g) ) clearly lift to invariant (holomorphic) sections upstairs (on C g ); conversely, we claim that invariant sections upstairs descend on the complement of a Zariski closed set of codimension 2 in C (g) , and then they extend throughout by virtue of Hartogs' Theorem.
Our claim follows because on an open set of the ramification locus the pull back divisor is trivial, and invariant functions are pull-backs of functions on the quotient.
For the last inclusion, we simply use that Remark 4.7.The graded ring A has the property that its subring A even is the graded ring associated to the pull-back ) is a representation of the group G of the Hypercube, hence R is a subring of A Sg , and it can be detected by considering the subring of sections of degree n vanishing of order n on the Diagonal ∆ ′ C , as done for instance by Canonaco in small genus [Can06].The best way to describe A even is to write its direct image on (P 1 ) g , but we do not pursue this further here.

Étale double covers of Hyperelliptic curves and Jacobians
Let ϕ : C ′ → C be an étale double covering of a Hyperelliptic curve C of genus g, so that Since the hyperelliptic involution ι acts trivially on Pic(C)[2], ι lifts to C ′ and we have an action of (Z/2) 2 on C ′ with quotient P 1 , that is, a bidouble cover of P 1 .
Hence (see [Cat84]) there is a factorization of the homogeneous polynomial f of degree 2g + 2 whose equation is the equation for the branch locus of ψ, with factors of respective degrees 2d 1 , 2d 2 , with d 1 + d 2 = g + 1, and such that This is the Eigenspace decomposition according to the (nontrivial) characters of (Z/2) 2 , and we identify H 0 (O P 1 (d)) to its pull-back under ϕ.The formula clearly shows that C ′ is hyperelliptic if and only if some d i = 1.
We run now for C ′ a similar game to the one we played for C: The first quotient is étale, while if we divide by the Symmetric group S n , we get above is no longer étale, since its degree equals 2 n , but the fibre cardinality drops over the discriminant hypersurface (for instance, if y ′ , y ′′ → y, then only three divisors 2y ′ , 2y ′′ , y ′ + y ′′ map to the divisor 2y).
Then Λ is normalized by S n , and defining Xn dominates C (n) via an étale double covering.
Proof.By the factorization Xn → C (n) is étale outside of the discriminant ∆ C , and since C (n) is smooth, it suffices to show that the covering is quasi-étale, that is, étale outside of codimension 2.
Given an effective divisor i m i y i , where y ′ i , y ′′ i → y i , we have as inverse images the effective divisors i m ′ i y ′ i +m ′′ i y ′′ i with m i = m ′ i +m ′′ i .For m i = 1, there are two possibilities, for m i = 2, as already observed, we have three possibilities: 2y ′ i , 2y ′′ i , y ′ i + y ′′ i .Assume that m 1 = 2, and all others m i = 1: then we simply observe, that, writing the divisors as images of the n-tuples (y ′ 1 , y ′ 1 , . . .), (y ′′ 1 , y ′′ 1 , . . .), (y ′ 1 , y ′′ 1 , . . .), there is an element of Σ, namely the involution j 1 × j 2 × identity which sends the first element to the second, and the third to (y ′′ 1 , y ′ 1 , . . .), which is equivalent modulo the action of S n .Hence, over the set of divisors with i (m i − 1) = 1 (whose complement has codimension 2), the inverse image consists of two distinct points of Xn .
The previous construction is especially useful in two cases: n = g, where it provides an étale double covering of Jac(C), which is birational to C (g) , and for n = g − 1, where it provides the corresponding étale double covering of the Theta divisor Θ C , which is birational to C (g−1) .
In order to simplify the exposition, we recall the following Lemma, whose proof can be found in [Andr58] (Proposition 3, page 806).
Lemma 5.2.There is a natural isomorphism between the canonical system on the symmetric product of a curve and the exterior product of the canonical system of the curve C Proposition 5.3.Let X := Xg−1 be as in Proposition 5.1 the étale double covering of C (g−1) : then the canonical image of X ⊂ P g is a finite covering Ŵ of P g−1 via a linear projection P g P g−1 .In terms of the two integers then the canonical image Ŵ of X is birational to the double covering of P g−1 branched on the union of two hyperplanes H := H 1 + H 2 .
When d 1 , d 2 ≥ 2, Ŵ is not a double covering of P g−1 .
Proof.The canonical system of Xn pulls back to the Λ ⋊ S n -invariant part of the canonical system of (C ′ ) n , which is . By Andreotti's Lemma 5.2 the S n -invariance determines a subspace isomorphic to Λ n (H 0 (Ω 1 C ′ )).We use now the formula and replace v 1 by u, v 2 by w, so that uw = v, and denoting u(i) for the section u on the i-th copy of C ′ , and similarly for the other variables, }. Set now n = g − 1, and observe that, taking Q i = M i = 0, that is, taking the invariants for (Z/2) g−1 , we get the canonical system of C n .Taking the further subring of S g−1 -invariants, we get ) and this linear system, by Theorem 3.2 corresponds to the morphism C (g−1) → P g−1 .
The other sections s for which we are looking for must be eigenvectors for the group Z/2 = (Z/2) g−1 /Λ, and with nontrivial eigenvalue, hence they must be left invariant by S g−1 and each σ i should send them to −s.
The second property implies that for them P i ≡ 0, for all i.
Hence we get exactly one new element v * , corresponding to the symmetrization of where we let w i := w(i) . . . .Observe now that, if d 1 , d 2 ≥ 2, then Q i , M i ≡ 0, and it is complicated to calculate v 2 * .We can however say that v * is not an eigenvector for ((Z/2) 2 ) g−1 , and v 2 * as well, hence v 2 * is not a section of a line bundle on P g−1 .If instead d 1 = 1, then Q i ≡ 0, then d 2 = g, and v * equals the symmetrization of u and is a multiple of u * := u 1 . . .u g−1 .
Hence the canonical map of X factors through the double covering given by u ).Then we see that, setting z 1 , z 2 to be the roots of f 1 , and z 3 , . . .z 2g+2 to be the roots of f 2 , then u 2 * = h 1 • h 2 , where h i is the linear form on P g−1 whose zero set is the hyperplane H i corresponding to the symmetrization of the divisor {z i } × (P 1 ) g−2 .

Application to canonical maps of hypersurfaces in Abelian Varieties
Let A be an Abelian variety of dimension g, and let X ⊂ A be a smooth ample hypersurface in A such that the Chern class c 1 (X) of the divisor X is a polarization of type d := (d 1 , d 2 , . . ., d g ), so that the vector space H 0 (A, O A (X)) has dimension equal to the Pfaffian d The classical results of Lefschetz [Lef21] say that the rational map associated to H

and is an embedding of
By adjunction, the canonical sheaf of X is the restriction O X (X), so a natural generalization of Lefschetz' theorems is to ask about the behaviour of the canonical systems of such hypersurfaces X.This behaviour depends on the hypersurface X and not just on the polarization type only, as shown in [Cat-Schr02]: if we have a polarization of type (1, 1, 2) then the image Σ of the canonical map Φ X is in general a surface of degree 12 in P 3 , birational to X, while for the special case where X is the pull-back of the Theta divisor of a curve of genus 3, then the canonical map has degree 2, and Σ has degree 6.
The canonical map of such a hypersurface X is, via the following folklore Lemma, a mixture of the restriction of the Lefschetz map with the Gauss map of X, which is a morphism for X smooth by a theorem of Ziv Ran [Ran84].Lemma 6.1.Let X be an ample hypersurface of dimension n in an Abelian variety A, such that the class of X is a polarization of type d := (d 1 , d 2 , . . ., d n+1 ).
Hence first of all the canonical map is an embedding if H 0 (A, O A (X)) yields an embedding of A; secondly, since a projection of Φ X is the Gauss map of X, given by ( ∂θ 1 ∂z 1 , . . ., ∂θ 1 ∂zg ), follows that the canonical system |K X | is base-point-free and Φ X a finite morphism.This is the main Theorem of [Cat-Ces21]: Theorem 6.2.Let (A, X) be a general pair, consisting of a hypersurface X of dimension n = g − 1 in an Abelian variety A, such that the class of X is a polarization of type d Then the canonical map Φ X of X is birational onto its image Σ.
6.1.A new proof of theorem 6.2.For the reader's convenience we borrow now a simple argument contained in [Cat-Ces21], yielding first a reduction step: Step I : It suffices to prove the Theorem in the case of a polarization of type (1, . . ., 1, p), with p a prime number, and assuming g ≥ 2.
We deal then with the following specializations: Consider now the respective ramification divisors R f , R = R Z , R W of the respective maps f : X → P , Ψ : Z → P , W → P .
Since X → Z is quasi-étale (unramified in codimension 1), R f is the inverse image of R, hence R f maps to R with mapping degree 2p.
We use now the notation and the results of Proposition 3.3.It turns out that all the components of R f have multiplicity 1, since the same happens for the components of R.
This excludes right away the case p ≥ 3, since a cyclic covering of degree p has a ramification divisor occurring with multiplicity (p − 1), and moreover P and Z do not admit unramified coverings.
Moreover, by Step II, we may assume that Z = Z, hence that g ≥ 4.
We are then left with the case where p = 2.
Here we can use Proposition 5.1, first under the assumption that we choose The image of X, which equals the one of X, is the double cover Ŵ of P with branch locus H 1 + H 2 , the union of 2 Hyperplanes.
Since W = Ŵ case (4) is clearly excluded, since Ŵ is not a double covering of Z.
Use now Proposition 5.1 under the assumption that d 1 , d 2 ≥ 2. Then case (3), where W would be the double covering of P = P g−1 branched on a branch divisor B ′ ⊂ ∆ ∪ H, is also excluded.
Step IV: To finish the proof, consider the more general case where X is a double étale covering of a smooth Theta divisor Θ.
Assume that we have a further factorization X → X/ι → Σ of the canonical map, and recall that X/ι is a double covering of Z.
Then, specializing to the case where we have the double étale covering X 0 of the Theta divisor of a hyperelliptic curve, we see by Theorem 2.7 that we have a further factorization of the canonical map of X 0 , X 0 → Σ 0 → Ŵ .
Indeed, Hypothesis (**) can be seen to hold using Theorem 4.3 and assertion (v) of Proposition 2.2.
As we argued before, Σ 0 is a double cover of Σ 0 /(Z/2), which is therefore either Z or Z.
In the latter case, if we take the degrees d 1 , d 2 ≥ 2, it would follow that Σ 0 = Ŵ .
Hence the monodromy of Σ → P would land in S 4 .However, if we specialize to the Theta divisor of a non hyperelliptic curve, the Monodromy group of the covering C (g−1) ∼ Θ C → P is equal to S 2g−2 , acting on the subsets of cardinality (g − 1).Since g ≥ 4, 2g − 2 ≥ 6 and the group A 2g−2 is simple: hence the monodromy image in S 4 has order 2, and cannot be transitive, whence a contradiction.
6.2.Final step.We repeat here the final argument which takes care of the case Σ = X/ι, as in [Cat-Schr02]: if for a general deformation of X as a symmetric divisor the canonical map would factor through ι, then X would be ι-invariant; being symmetric, it would be (Z/2) 2 -invariant, hence for all deformations X would remain the pull-back of a Theta divisor.This is a contradiction, since the Kuranishi family of X has higher dimension than the Kuranishi family of a Theta divisor Θ (see ).
στ.Then assertion (i) follows from the Claim: The only S n -invariant subspaces of K are : {0}, K, (Z/2)e, e ⊥ , where e := n 1 e i .