The Prym Map: from Coverings to Abelian Varieties

The Prym Map: From Coverings to Abelian Varieties

Angela Ortega

Introduction projective variety. Although it can be defined over an arbi- Abelian varieties are very important objects that lie at the trary field 퐾, in this note we will only consider the case of crossroads of several areas in mathematics: algebraic ge- complex abelian varieties, that is 퐾 = ℂ. ometry, complex analysis, and number theory. So they Abelian varieties made their appearance at the begin- can be studied from different points of view. An abelian ning of the nineteenth century, when Niels Henrik Abel variety over a field 퐾 is an abelian group which admits and Carl Gustav Jacob Jacobi studied hyperelliptic integrals. an embedding in a projective space ℙ(퐾), so it is also a A hyperelliptic integral is an integral of the form 푝(푧) Angela Ortega is a researcher in mathematics at Humboldt Universität zu ∫ 푑푧, Berlin. Her email address is [email protected]. 훾 √푞(푧) Communicated by Notices Associate Editors Daniel Krashen and Steven Sam. where 푝(푧) is a polynomial, 푞(푧) = (푧 − 푎1) ⋯ (푧 − 푎푛) is a For permission to reprint this article, please contact: polynomial of degree 푛 with pairwise distinct roots 푎푖, and [email protected]. 훾 is a path in ℂ. At the time of Abel and Jacobi it was known DOI: https://doi.org/10.1090/noti2150 that for 푛 = 1, 2 this integral could be solved by means

1316 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 of trigonometric and logarithmic functions. For 푛 = 3, 4 one can solve it using elliptic functions, but for 푛 ≥ 5 no way of integrating this function is known in general. Con- 1 sidered as a function on ℂ, the differential 푑푧 is not √푞(푧) singled-valued, but it can be seen as a holomorphic differential on the compact Riemann surface (which is also a √ line bundle 퐿 on the torus 푉/Γ, called a polarization. The smooth algebraic curve over ℂ) associated to 푞(푧), that is, ⊗푘 1 sections of 퐿 , for some 푘 > 0, provide an embedding of the double covering 퐶 of ℙ ramified at the points 푎1, … , 푎푛. 1 0 ⊗푘 ∗ 푁 1 퐴 into a projective space ℙ퐻 (퐴, 퐿 ) ≃ ℙ : The difficulty to integrate 푑푧 comes from the topolog- √푞(푧) 푥 ↦ [푠 (푥) ∶ 푠 (푥) ∶ ⋯ ∶ 푠 (푥)], ical structure of 퐶. Abel and Jacobi had the idea to inte- 0 1 푁 grate all the holomorphic differentials on 퐶 at the same where 푠0, 푠1, … , 푠푁 form a basis of the space of sections 푖−1 1 퐻0(퐴, 퐿⊗푘). A polarization has several incarnations: (1) time by considering a basis of the form 휔푖 ∶= 푧 푑푧, √푞(푧) 2 푛−1 푛−1 an ample line bundle 퐿; (2) a nondegenerated alternating 푖 = 1, … , [ ]. The dimension 푔 ∶= [ ] of the vector 2 2 bilinear form 퐸 ∶ 푉 × 푉 → ℝ with 퐸(Γ, Γ) ⊂ ℤ; (3) a non- 0 space of holomorphic differentials (denoted by 퐻 (퐶, 휔퐶)) degenerated Hermitian form 퐻 on 푉 with Im 퐻(Γ, Γ) ⊂ ℤ; is called the genus of 퐶. Topologically, the genus is the and (4) a Weil divisor Θ ⊂ 푋 with the property that the number of “holes” of the Riemann surface 퐶 (see Figure set {푎 ∈ 퐴 ∣ Θ − 푎 ∼ Θ} is finite. One has 퐿 = 풪퐴(Θ), 1). Let 푝0 ∈ 퐶 be a fixed point. Around this point wecan the associated line bundle to the divisor Θ; conversely, giv- define the map ing a polarization 퐿 one recovers Θ as the zero locus of a 푝 푝 section of 퐿. There is a basis of 푉 with respect to which 0 퐷 푝 ↦ (∫ 푤1,…, ∫ 푤푔) , (1) the alternating form 퐸 has as matrix ( −퐷 0 ), where 퐷 is a 푝0 푝0 diagonal matrix with positive integer entries 푑1, … , 푑푔 sat- but it cannot be extended to the whole 퐶 because these in- isfying 푑푖|푑푖+1 for 푖 = 1, … , 푔 − 1. The vector (푑1, … , 푑푔) is 퐿 tegrals depend on the path between 푝0 and 푝. The way called the type of the polarization of , and when it is of the out is to consider the image modulo the values of inte- form (1, … , 1) the polarization is principal. In this case the grals along closed paths to obtain a well-defined map. Let dimension of the space of sections of 퐿 is one, so we have 2푔 퐻1(퐶, ℤ) ≃ ℤ be the group of closed paths in 퐶 (which canonically associated to the abelian variety a unique geo- does not depend on the starting point) modulo homol- metric object, its theta divisor. This is one of the reasons ogy. This group can be seen as a full rank lattice inside why principal polarizations are the most studied ones. 0 ∗ 퐶 퐻 (퐶, 휔퐶) , via the injective map As we have seen, the Jacobian of an algebraic curve (or compact Riemann surface) is the complex torus 0 ∗ 훾 ↦ {휔 ↦ ∫휔} 퐽퐶 = 퐻 (퐶, 휔퐶) /퐻1(퐶, ℤ). 훾 The intersection product on 퐻1(퐶, ℤ) induces an alternat- assigning to a path 훾 the functional which integrates the 0 ∗ ing form 퐸 on 푉 ∶= 퐻 (퐶, 휔퐶) . More precisely, if we holomorphic differentials along 훾. Thus, the map (1) ex- choose a basis over ℤ, 훾1, ..., 훾2푔 of 퐻1(퐶, ℤ) as in Figure tends to a holomorphic map on 퐶, 0 ퟏ 1, the intersection product has as matrix ( 푔 ). As 푝 푝 −ퟏ푔 0 퐻 (퐶, ℤ) is a full rank lattice in 푉, the {훾 } form also a basis 훼 ∶ 푝 ↦ (∫ 푤1,…, ∫ 푤푔) mod 퐻1(퐶, ℤ), 1 푖 푝0 푝0 of 푉 as ℝ-vector space. One verifies then that with respect called the Abel–Jacobi map, and the complex torus 퐽퐶 ∶= to this basis, the intersection matrix gives an alternating 0 ∗ form 퐸 on 푉 defining a principal polarization Θ. 퐻 (퐶, 휔퐶) /퐻1(퐶, ℤ) is the Jacobian variety of 퐶. Conse- quently, the problem of integrating holomorphic differen- A one-dimensional abelian variety is also an algebraic tials on 퐶 is equivalent to describing the Jacobian variety curve of genus one, that is, an elliptic curve. The Jacobian of 퐶 and the Abel–Jacobi map 훼 ∶ 퐶 → 퐽퐶. Historically, of a genus one curve is then isomorphic to the curve itself. the Jacobian of a compact Riemann surface 퐶 is the first Algebraic geometers typically gather their objects of example of an abelian variety. study in families to investigate a general property or to The main idea we want to convey in this article is that single out interesting elements. Ideally, the set of all the one can construct and study abelian varieties using curves objects is itself an algebraic variety where one can apply and its surrounding geometry. known tools. This leads to the notion of moduli space, which is the variety parametrizing the objects. Fortunately, Abelian varieties and Jacobians. One can show that a complex abelian variety 퐴 is the quotient of a complex vector 1This is essentially the definition of ampleness of a line bundle. space 푉 by a full rank lattice Γ in 푉 together with an ample 2Or more precisely its first Chern class.

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1317 Since much more is known about the moduli space of curves than the moduli of abelian varieties, the answer to this question will help us understand better abelian varieties and their moduli spaces. Coverings of curves and Prym varieties. Consider a finite covering 푓 ∶ 퐶˜ → 퐶 of degree 푑 between two smooth projective curves, and let 푔 and ̃푔 denote the genera of 퐶 and 퐶˜, respectively. By the Riemann–Hurwitz formula these Figure 1. Curve of genus 푔. genera are related by deg 푅 there exists a nice parameter space for all principally polar- ̃푔= 푑(푔 − 1) + + 1, (2) 2 ized abelian varieties (ppav) of fixed dimension 푔 (up to where 푅 denotes the ramification divisor of 푓, that is, the isomorphism). This moduli space can be constructed as set of points in 퐶˜ (counted with multiplicities) where the the quotient of the Siegel upper half plane map is not locally a homeomorphism. The map 푓 induces 푡 픥푔 ∶= {휏 ∈ 푀푔×푔(ℂ) ∣ 휏 = 휏, Im 휏 > 0} a map between the Jacobians of the curves, the norm map. As a group, the Jacobian 퐽퐶 is generated by the points of by the action of the symplectic group the curve 퐶, and in fact 퐽퐶 parametrizes classes of linear 푆푝2푔(ℤ) equivalence of divisors of degree zero. With this in mind 0 ퟏ퐠 푡 0 ퟏ퐠 one can simply define the norm map as the pushforward = {푀 ∈ 퐺퐿2푔(ℤ) ∶ 푀 ( ) 푀 = ( )} −ퟏ퐠 0 −ퟏ퐠 0 of divisors from 퐶˜ to 퐶: given by Nm푓 ∶ 퐽퐶˜ → 퐽퐶, [∑ 푛푖푝푖] ↦ [∑ 푛푖푓(푝푖)] , 푖 푖 푎 푏 −1 푀 = ( ) ∈ 푆푝2푔(ℤ), 푀 ⋅ 휏 = (푎 + 푏휏)(푐 + 푑휏) . 푐 푑 where the sum is finite, ∑ 푛푖 = 0 with 푛푖 ∈ ℤ, and the Thus, every point in the quotient 픥 /푆푝 (ℤ) represents an bracket denotes the class of linear equivalence. The ker- 푔 푔 Nm Nm isomorphism class of a principally polarized abelian vari- nel of 푓 is not necessarily connected, but since 푓 ety of dimension 푔: for each 휏 ∈ 픥 set 퐴 = ℂ푔/휏ℤ푔 ⊕ ℤ푔, is a homomorphism of groups the connected component 푔 휏 ˜ then containing the zero is naturally a subgroup of 퐽퐶. This subgroup is the Prym variety of 푓 denoted by 퐴 ≃ 퐴 ′ as ppav 휏 휏 푃(푓) ∶= (Ker Nm )0 ⊂ 퐽퐶.˜ (3) ′ 푓 ⇔ ∃ 푀 ∈ 푆푝2푔(ℤ) s.t. 휏 = 푀 ⋅ 휏. Moreover, the restriction of the principal polarization Θ In what follows, we denote by 풜푔 the moduli space of on 퐽퐶˜ to 푃(푓) defines a polarization Ξ; hence (푃(푓), Ξ) is principally polarized abelian varieties of dimension 푔. Ob- an abelian subvariety of the Jacobian 퐽퐶˜ of dimension serve that the dimension of this space is the same as the di- dim 푃(푓) = dim 퐽퐶˜ − dim 퐽퐶 = ̃푔− 푔. mension of the space of symmetric matrices of size 푔; thus 푔(푔+1) The Prym variety can be regarded as the complementary dim 퐴푔 = . 2 variety of the image of 푓∗ ∶ 퐽퐶 → 퐽퐶˜ inside 퐽퐶˜. It was The space parametrizing isomorphism classes of shown by Mumford in [Mum74] that the restricted po- smooth projective curves has no such simple description as larization Ξ is principal when 푓 is an étale(unramifed) the moduli of abelian varieties, but it is nevertheless a well- double covering, or when it is a double covering ramified behaved algebraic variety. Let ℳ be the moduli space of 푔 in exactly two points. These are basically the only cases smooth projective curves of genus 푔. It is an irreducible al- when one can obtain a principal polarization on 푃(푓). As- gebraic variety of dimension 3푔 − 3. By associating to each sume now that 푓 is an étaledouble covering. According smooth curve [퐶] ∈ ℳ its Jacobian, we get the Torelli map: 푔 to (2) the dimension of the corresponding Prym variety is 픱 ∶ ℳ푔 → 풜푔, [퐶] ↦ (퐽퐶, Θ). dim 푃(푓) = 2(푔 − 1) − 푔 = 푔 − 1. Thus, this construction provides us a way to associate to each étaledouble covering The celebrated Torelli theorem states that the map 픱 is an em- 푓 ∶ 퐶˜ → 퐶 over a smooth curve 퐶 of genus 푔 a principally bedding. In particular, one can recover a smooth algebraic polarized abelian variety; this is the Prym map. In order curve 퐶 from the pair (퐽퐶, Θ). Comparing the dimensions to make the definition precise we need to introduce the of both spaces, one deduces that the general principally moduli space polarized abelian variety of dimension < 3 is the Jacobian of some curve. ℛ푔 ∶= {[퐶, 휂] ∣ [퐶] ∈ ℳ푔, 0 ⊗2 Which other abelian varieties can be obtained from curves? 휂 ∈ Pic (퐶) ⧵ {풪퐶}, 휂 ≃ 풪퐶}/ ≅

1318 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 parametrizing all the (nontrivial) ´etaledouble coverings The Prym Map 푃푟6 ∶ ℛ6 → 풜5 over curves of genus 푔 up to isomorphism. Given a pair The case of ´etaledouble coverings over a genus 6 curve de- [퐶, 휂] ∈ ℛ the isomorphism 휂⊗2 ≃ 풪 endows 풪 ⊕ 푔 퐶 퐶 serves special attention. We have dim ℛ6 = dim 풜5 = 15, 휂 with a ring structure (actually with a structure of 풪 - 퐶 and the map 푃푟6 being dominant implies that it is also algebra). Thus, the corresponding double covering is given generically finite. Can one determine the degree oreven ˜ by taking the spectrum 퐶 ∶= Spec(풪퐶 ⊕ 휂), and the map 푓 describe its generic fibre? There is not only a positive an- is just the natural projection Spec(풪퐶 ⊕휂) → 퐶 = Spec 풪퐶, swer to this question but also a beautiful one. The degree 풪 ↪ 풪 ⊕ 휂 induced by the inclusion 퐶 . There are finitely of 푃푟6 ∶ ℛ6 → 풜5 is 27, which is also the number of lines many “square roots” of 풪퐶, that is, line bundles 휂 with on every smooth cubic surface. This is not a coincidence, ⊗2 휂 ≃ 풪퐶. More precisely, the forgetful map because the monodromy group of the Prym map equals the Weyl group 푊(퐸 ), which governs the incidence struc- ℛ → ℳ , [퐶, 휂] ↦ 휂 6 푔 푔 ture of the lines in a smooth cubic surface ([Don92]). On 2푔 the other hand, 푃푟6 fails to be finite over the locus of all is finite of degree 2 − 1, and hence dim ℛ푔 = dim 푀푔 = 3푔 − 3. The Prym map is then defined as the Jacobians of curves of genus 5, as well as over the locus of intermediate Jacobians of cubic threefolds.3 The fibres 푃푟푔 ∶ ℛ푔 → 풜푔−1 [퐶, 휂] ↦ (푃(푓), Ξ). and the blowup of the Prym map along these loci are ex- plicitly described in [Don92]. There is even a procedure to By comparing the dimensions on both sides, one sees that pass from one element to another on a general fibre of 푃푟6: 푔(푔−1) dim ℛ푔 ≥ dim 풜푔−1 = for 푔 ≤ 6, so it makes sense the tetragonal construction. First, notice that a general curve 2 1 to ask if for low values of 푔 the Prym map is dominant, i.e., [퐶] ∈ ℳ6 carries exactly five 푔4’s, that is, line bundles of if we can realize a (general) principally polarized abelian degree 4 on 퐶 whose space of sections is two-dimensional. 1 1 variety of dimension ≤ 6 as the Prym variety of some cover- Thus, a 푔4 on 퐶 is equivalent to having a 4:1 map 퐶 → ℙ . ˜ ing. The following theorem, representing work of several Now, given a double covering [푓 ∶ 퐶 → 퐶] ∈ ℛ6 and a 1 mathematicians, gives a positive answer to this question 푔4 on 퐶, one can construct two other coverings in ℛ6 as and summarizes the situation for the classical Prym map. follows. Let 퐶(4) (respectively, 퐶˜(4)) be the fourth symmetric product of 퐶 (respectively, 퐶˜), parametrizing divisors Theorem 4. (a) The Prym map is dominant if 푔 ≤ 6. of degree 4 on the corresponding curves. Define the curve (b) The Prym map is generically injective if 푔 ≥ 7. 푋˜ by the cartesian diagram (c) The Prym map is never injective. 푋˜ / 퐶˜(4) Let 풫푔−1 denote the image of 푃푟푔. Wirtinger showed (4) 16∶1 푓 4 푓(4) ([Wir95]) that the closure 풫 is an irreducible subvari- |푋˜ 2 ∶1 푔−1 1 1 / (4) ety in 풜푔−1 of dimension 3푔 − 3, so 풫푔−1 = 풜푔−1 for 푔 ≤ 6, ℙ = 푔4 퐶 which implies part (a). Moreover, he also proved that the (4) Jacobian locus in 풜푔−1 (i.e., the image of the Torelli map where 푓 is the natural map 푝1 + ⋯ + 푝4 ↦ 푓(푝1) + ˜ 픱) is contained in 풫 . In this sense, Pryms are a general- ⋯ + 푓(푝4). The involution 휎 on 퐶 induces an involution 푔−1 ˜ ization of Jacobians. Part (b) was first proved by R. Fried- ̃휎∶ 푝1 +⋯+푝4 ↦ 휎(푝1)+⋯+휎(푝4) on 푋. It turns out that ˜ man and R. Smith ([FS82]), and for 푔 ≥ 8 by V. Kanev 푋 consists of two disjoint nonsingular connected compo- ˜ ˜ ([Kan82]) by using degeneration methods. More geomet- nents 푋0, 푋1 and the involution ̃휎 acts without fixed points ric proofs were given by G. Welters ([Wel87]) and later by on each of these components. Moreover, the restriction of (4) ˜ ˜ O. Debarre ([Deb89]), in the spirit of the proof of Torelli’s the map 푓 to 푋0 and 푋1 defines 8:1 maps fitting inthe theorem. The fact that the Prym map is noninjective was following diagram: first observed by Beauville ([Bea77]), who produced, using 푋˜ ⊔ 푋˜ Recillas’ trigonal construction, examples of nonisomorphic 00 1 ?? ~~ 00 ? 푓0 ~~ 0 ??푓1 coverings (퐶˜1, 퐶), (퐶˜2, 퐶2) in ℛ푔 for 푔 ≤ 10, whose Prym va- ~ 0 ? ~~ 0 ?? rieties are isomorphic as principally polarized abelian va- ~ 00 08∶1 8∶1 푋 PP 0 n 푋 rieties. Later, Donagi’s tetragonal construction ([Don81]) 0 PPP 0 nnn 1 PP 00 nn (of which Recillas’s construction is a degeneration) pro- P4∶1PP 0 4∶1nnn PPP 0 nnn vided examples for the noninjectivity of the Prym map in PP' ×wnnn any genus. We will discuss this construction in the next ℙ1 section. When 푔 < 6 the fibers of the Prym map are positive 3The intermediate of a cubic threefold 푌 is the complex torus 1,2 dimensional and they are geometrically well understood. 퐻 (푌)/퐻3(푌, ℤ).

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1319 where 푋푖 ∶= 푋˜푖/ ̃휎 for 푖 = 0, 1. Therefore, from an element automorphism. If we denote by the same letter the auto- 푓 morphism on 퐽퐶˜ induced by 휎, it is more convenient to [퐶˜ → 퐶] ∈ ℛ together with a pencil 푔1 on 퐶 we have con- 6 4 define the Prym variety of the covering 푓 as 푓 ℛ [푋˜ →0 푋 ] structed two other elements in 6, namely, 0 0 and 푃(푓) ∶= Im(1 − 휎) ⊂ 퐽퐶,˜ 푓 [푋˜ →1 푋 ] 푔1 푋 푋 1 1 , together with 4’s on 0 and 1. These three which is an abelian subvariety of 퐽퐶˜ of dimension pairs are in general not isomorphic, but the associated Pryms are isomorphic ([Don81]). If one applies the con- dim 푃(푓) = ̃푔− 푔 = (푑 − 1)(푔 − 1) + 푟 =∶ 푝, 푓0 struction to [푋˜ → 푋 ] using the obtained 푔1, one gets back with polarization Ξ given by the restriction of the prin- 0 0 4 ̃ ˜ the original two coverings, but if one uses another pencil cipal polarization Θ on 퐽퐶. It is a polarization of type 퐷 = (1, … , 1, 푑, … , 푑), where 1 occurs 푝 − (푔 − 1) times and on 푋0, one gets two other nonisomorphic coverings. By repeating this procedure on the coverings using different 푑 occurs 푔−1 times if 푟 = 0, and 1 occurs 푝−푔 times and 푑 pencils on their base curves, we obtain eventually all the occurs 푔 times if 푟 > 0. This definition coincides with (3), 0 elements on the fibre. This is also the structure on the lines since in that case Im(1 − 휎) = ker(1 + 휎) . Let of a smooth cubic surface. Two coverings in the fibre are ℛ푔(푑, 푟) ∶= {(퐶, 퐵, 휂) ∣ related by the tetragonal construction, as two lines on a cu- 휂 ∈ Pic푚(퐶), 퐵 reduced divisor in |휂⊗푑|}/ ≅ bic surface are incident. The triad {(퐶,˜ 퐶), (푋˜ , 푋 ), (푋˜ , 푋 )} 0 0 1 1 denote the moduli space parametrizing cyclic coverings of corresponds to a triangle in the surface. The fact that each degree 푑 over a curve of genus 푔 totally ramified over 푑푚 line in the cubic surface is a “side” of five triangles is re- points, and let 풜퐷 be the moduli space of abelian varieties flected in the construction by the five 푔1’s that a curve 퐶 푝 4 of dimension 푝 and polarization type 퐷. In this case the possesses. This particular case shows that the fibres of the Prym map is given by Pym map display rich and beautiful geometry. 푓 Nonprincipally polarized abelian varieties. One can of 퐷 푃푟푔(푑, 푟) ∶ ℛ푔(푑, 푟) → 풜푝 ,[퐶˜ → 퐶] ↦ (푃(푓), Ξ). course use more general coverings to construct abelian varieties, for instance, coverings of degree higher than 2 or In order to compute the dimension of its image one needs ramified ones. In these cases one obtains abelian varieties to know when this map is generically finite. This is the which are no longer principally polarized. In this section case when the differential map of 푃푟푔(푑, 푟) is injective at a 2푟 generic point of ℛ푔(푑, 푟) or equivalently when the codiffer- we will discuss cyclic coverings ramified at points with ∗ 푟 ≥ 0. ential map 푑 푃푟푔(푑, 푟) is surjective. One of the advantages By definition, an ´etalecyclic covering 퐶˜ → 퐶 of de- of considering cyclic covering is that the tangent space at gree 푑 admits an automorphism 휎 of order 푑 on 퐶˜ such 0 ∈ 푃(푓) to the Prym variety can be identified with the that 퐶 = 퐶/⟨휎⟩˜ . For simplicity, we can assume that 푑 direct sum of spaces of sections is a prime number (otherwise one can factorize the map 푑−1 푇 푃 ≃ 퐻0(퐶, 휔 ⊗ 휂푖)∗, through cyclic coverings of smaller degree). If the cyclic 0 ⨁ 퐶 covering is branched over a divisor 퐵 ⊂ 퐶, the ramifica- 푖=1 tion index at each ramified point is 푑; i.e., all the branches where each summand is an eigenspace for the action of 휎 deg 퐵 = 푑푚 0 ˜ ∗ come together in that point. In particular, if , on 퐻 (퐶, 휔퐶˜) . Notice that the forgetful map [퐶, 퐵, 휂] ↦ for some integer 푚 > 0, the ramification divisor is of de- [퐶, 퐵] is finite over the moduli space ℳ푔,푑푚 of 푑푚-pointed gree smooth curves of genus 푔. Therefore the cotangent space deg 푅 = 2푟 = (푑 − 1)푑푚. to a generic point [퐶, 휂, 퐵] ∈ ℛ푔(푑, 푟) can be identified to Similarly to the ´etaledouble cover case, giving a ramified the cotangent space to ℳ푔,푑푚 at [퐶, 퐵]. By identifying the cyclic covering is equivalent to giving a triple (퐶, 퐵, 휂) with cotangent spaces ⊗푑 2 휂 a line bundle on 퐶 (of degree 푚) satisfying 휂 ≃ 풪퐶(퐵). ∗ 퐷 ∗ 푇(푃,Ξ)풜푝 ≃ Sym (푇0푃) , More precisely, take a section 푠 of 풪 (퐵) vanishing exactly 퐶 푇∗ ℛ (푑, 푟) ≃ 퐻0(퐶, 휔 (퐵)), along 퐵 (if 퐵 = ∅, take 푠 the constant section 1). Denote [퐶,휂,퐵] 푔 퐶 by 푇표푡(휂) the total space of 휂 and let 푝 ∶ 푇표푡(휂) → 퐶 be we obtain that the codifferential of 푃푔(푑, 푟) at a generic the projection. If 푡 is the tautological section of the line point [퐶, 퐵, 휂] is given by the multiplication of sections bundle 푝∗휂 → 푇표푡(휂), then the locus where the section 푑∗푃푟 (푑, 푟) ∶ Sym2(푇 푃)∗ → 퐻0(퐶, 휔2 ⊗ 풪(퐵)). 푝∗푠 − 푡푑 vanishes defines the curve 퐶˜ inside 푇표푡(휂). 푔 0 퐶 Recall that points in the Jacobian represent line bundles In the cases when this map is surjective at the generic point of degree zero on the corresponding curve. Hence, every [(퐶, 퐵, 휂)] we get that the Prym map 푃푟푔(푑, 푟) is generically automorphism of a curve induces an automorphism of its finite onto its image. Recall that 푑푚 is the degree of the Jacobian by taking the pullback of the bundle under the branch locus of the covering.

1320 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 Theorem 1 ([LO11]). If endomorphism of 퐽퐶. A nontrivial endomorphism 훾퐷 on • 푔 ≥ 2 and 푑푚 ≥ 6 for 푑 even or 푑푚 ≥ 7 for 푑 odd, 퐽퐶 defines a proper subvariety 푃 ∶= Im(1 − 훾퐷) of the Jaco- • 푔 ≥ 3 and (푑, 푚) ∈ {(4, 1), (5, 1), (2, 2), (3, 2)}, bian. • 푔 ≥ 5 and (푑, 푚) ∈ {(2, 1), (3, 1)}, We say that a principally polarized abelian variety (푃, Ξ) is a Prym–Tyurin variety4 for the curve 퐶 of exponent 푒 if then the Prym map 푃푟푔(푑, 푟) is generically finite. 휄 푃 ↪ 퐽퐶 is a subvariety and 휄∗Θ ≡ 푒Ξ. One checks that In the case of ramified double coverings we know that in this case the endomorphism Norm 푁푃 ∈ End 퐽퐶 associ- the Prym map is generically injective as soon as the dimen- ated to 푃 satisfies the equation 푁푃(푒−푁푃) = 0. For example, sions of the moduli spaces in the source and target allow it Jacobians are Prym–Tyurin varieties of exponent one, and (see [MP12], [MN14], [NO19]), except when 푔 = 3, 푟 = 2, classical Prym varieties are Prym–Tyurin of exponent two. where deg 푃푟3(2, 2) = 3. There are actually at least two dif- Welters ([BL04, Corollary 12.2.4]) proved that every princi- ferent ways to interpret this degree ([BCV95], [NR95]). As pally polarized abelian variety of dimension 푔 is in fact a we have seen in the previous section, it is particularly ap- Prym–Tyurin variety for some curve of a very large genus, pealing to study the geometry of the fibres of the Prym with exponent 3푔−1(푔 − 1)!. It is an open problem to find, 퐷 map. Let ℬ퐷 be the component of the moduli space 풜푝 for a given 푔, the smallest integer 푐 such that every princi- of elements (푃, Ξ) such that the polarization Ξ is compat- pally polarized abelian variety of dimension 푔 is a Prym– ∗ 퐷 ible with 휎, i.e., 휎 Ξ ≡ Ξ. In particular, ℬ퐷 = 풜푝 when Tyurin variety of exponent 푒 ≤ 푐. The issue is that, as with 푑 = 2. One checks that Im 푃푟푔(푑, 푟) ⊂ ℬ퐷. By computing automorphisms, the general curve does not carry a non- the dimension of ℬ퐷 one can prove that only in the follow- trivial correspondence, which leaves us with little room to ing cases is 푃푟푔(푑, 푟) generically finite and dominant over look for Prym–Tyurin varieties. However, many examples ℬ퐷 ([LO18]): of families of Prym–Tyurin varieties have been constructed by employing different tools like representation theory or (푔, 푑, 푟) deg 푃푟푔(푑, 푟) dim 푃 Polarization the geometry of lines in some Fano varieties. (6,2,0) 27 5 (2,2,2,2,2) The following Prym–Tyurin map, worked out in [ADF+] (3,2,2) 3 4 (1,2,2,2) after the original construction of V. Kanev ([Kan89]), gives (1,2,3) 1 3 (1,1,2) us a parametrization of 풜6 through special coverings; it (4,3,0) 16 6 (1,1,1,3,3,3) illustrates how to use geometric objects to produce Prym– (2,7,0) 10 6 (1,1,1,1,1,7) Tyurin varieties. Consider a smooth cubic threefold 푉 in 4 4 (2,3,3) ? 5 (1,1,1,3,3) ℙ and choose a pencil of hyperplanes {퐻휆}휆∈ℙ1 in ℙ , that is, a family of hyperplanes parametrized by ℙ1. With the The degree of the Prym map 푃푟2(3, 3) seems to be un- right choice of the pencil, the intersection 푉휆 ∶= 퐻휆 ∩ 푉 is known. We would like to emphasize that the fibre of the a smooth cubic surface for the general element of the family. different Prym maps carries a peculiar structure, so the way Actually, one can compute that there are exactly 24 critical of computing the degree has been ad hoc for each case. values of 휆 for which this intersection is a singular cubic Beyond coverings: Prym–Tyurin varieties. In the previ- surface. On these singular surfaces some of the lines come ous construction the Prym variety appears as a subvariety together, so they contain 21 lines instead of 27. We define of some Jacobian. It is known that to every abelian subva- the curve of all the lines in the family: riety of a Jacobian corresponds an endomorphism, called 퐶 ∶= {(휆, ℓ) | ℓ a line in 푉 }, the norm-endomorphism. Thus, in order to extend the defi- 휆 nition of Prym variety, one needs to look at Jacobians ad- which is a smooth curve in the Grassmannian 픾(1, 4) of mitting nontrivial endomorphisms, i.e., endomorphisms lines in ℙ4. The projection on the first factor defines afi- which are not of the form 푎 ↦ 푛 ⋅ 푎, with 푛 ∈ ℤ (ev- nite morphism ℎ ∶ 퐶 → ℙ1 of degree 27, since a smooth ery abelian variety contains ℤ as a subgroup of endomor- cubic surface contains 27 lines. Moreover, over each of phisms). As we mentioned before, every automorphism of the 24 critical points the map ℎ has 6 simple ramification the curve induces an automorphism of the Jacobian, and points, corresponding to the places where 2 lines collide more generally, every endomorphism of 퐽퐶 comes from a giving rise to a singular cubic surface. By the Riemann– correspondence on the curve. A correspondence on 퐶 is a Hurwitz formula it is easy to compute that 퐶 has genus divisor 퐷 ⊂ 퐶 × 퐶, which induces a map from the curve to 46. Now, we will use the incidence of the lines to define a the group of divisors on 퐶 of fixed degree: correspondence on 퐶: 푛 퐶 → Div (퐶), 푥 ↦ 퐷| . ′ ′ {푥}×퐶 퐷 ∶= {((휆, ℓ), (휆, ℓ )) ∈ 퐶 × 퐶 | ℓ ∩ ℓ ≠ ∅ in 푉휆} Since the curve 퐶 generates 퐽퐶 as a group, by the univer- sal property of the Jacobian we can extend this map to an 4Also called generalized Prym variety.

OCTOBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1321 and we denote by 퐷(ℓ) the divisor of degree 10 in 퐶 corre- denotes the induced endomorphism on 퐽퐶 we define the sponding to the lines which are incident to ℓ. The geome- associated Prym–Tyurin variety try of the lines in a cubic surface imposes a relation (푃푇(퐶, 훾), Ξ) 1 퐷(퐷(ℓ)) + 4퐷(ℓ) − 5ℓ = 5 푔27 as the image of 훾 − 1 inside 퐽퐶, and since the correspon- as an equality of divisor classes on 퐶, where the right-hand dence has the desired properties, the induced polarization side is the divisor consisting of five times the fibre of ℓ. is indeed principal, so it defines an element in 풜6. One

This equality implies that the divisor 퐷 induces an endo- observes that ℋ퐸6 depends on morphism 훾퐷 on the Jacobian 퐽퐶, satisfying the quadratic #{branch points} − dim Aut(ℙ1) = 24 − 3 = 21 equation parameters. The following theorem confirms the intuition 2 훾퐷 + 4훾퐷 − 5 = (훾퐷 + 5)(훾퐷 − 1) = 0. + that ℋ퐸6 is a parameter space for 풜6 ([ADF ]): It is not difficult to check that the correspondence has Theorem 2. The Prym–Tyurin map no fixed points (the diagonal in 퐶 × 퐶 does not inter- 푓 sect 퐷), and it is symmetric, that is, (휆, ℓ), (휆, ℓ′)) ∈ 퐷 ⇔ 1 푃푇 ∶ ℋ퐸6 → 풜6, [퐶 ⟶ ℙ ] ↦ (푃푇(퐶, 훾), Ξ) ((휆, ℓ), (휆, ℓ′)) ∈ 퐷. Then, by a criterion due to Kanev, one is generically finite. In particular, the general principally polar- can conclude that the abelian variety defined as ized abelian variety of dimension 6 is a Prym–Tyurin variety of 푃푇(퐶, 퐷) ∶= Im(훾퐷 − 1) ⊂ 퐽퐶 exponent 6 for a curve of genus 46. carries a principal polarization Ξ, more precisely Θ |푃푇(퐶,퐷) There are of course other spaces parametrizing 풜6 but ≡ 6Ξ. Therefore 푃푇(퐶, 퐷) is a Prym–Tyurin variety of expo- this one involves curves, generalizing somehow the classi- nent 6 inside 퐽퐶, and one computes that dim 푃푇(퐶, 퐷) = 6. cal Prym map for 푔 = 4, 5. A great deal is known about the This is a fortunate observation since the attempt to domi- geometry of Hurwitz schemes, so hopefully this construc- nate the moduli space 풜푔 using Prym varieties failed for tion will be useful to study 6-dimensional abelian varieties 푔 ≥ 6. In fact, a breakthrough theorem states that 풜 푔 and the birational nature of 풜6. Notice the big leap on the is of general type for 푔 ≥ 7 ([Mum83]), which in partic- complexity of the parameter space of 풜푔: for 푔 = 5 the ular means that these moduli spaces are as far as possi- group acting on the coverings is ℤ2, whereas for 푔 = 6 the ble from being parametrized by projective spaces. On the Jacobians of the coverings in ℋ퐸 admit an action of 푊(퐸6) 5 6 other hand, 풜푔 is unirational for 푔 ≤ 5. But how about which is of order 51,840. It is also shown in [ADF+] that 풜6? a compactification ℋ퐸6 of this Hurwitz space is of general By counting the parameters involved in the construc- type, which leaves us with the open question of whether tion one can verify that there are not enough Prym–Tyurin 풜6 is of general type or not. varieties constructed as above to dominate 풜6. The space parametrizing all cubic threefolds is of dimension 10, and References the dimension of the pencils of hyperplanes in ℙ4 is 6, so [ADF+] V. Alexeev, R. Donagi, G. Farkas, E. Izadi, and A. Or- this geometrical construction does not cover the moduli tega, The uniformization of the moduli space of principally po- space 풜6 which is of dimension 21. The key point is that larized abelian 6-folds, to appear in the Journal für die reine we do not need cubic threefolds to build up a curve 퐶 with und angewandte Mathematik. a correspondence of degree 10. What is essential is the ac- [BCV95] F. Bardelli, C. Ciliberto, and A. Verra, Curves of mini- mal genus on a general abelian variety, Compositio Math. (2) tion of the Weyl group 푊(퐸 ) on the lines of a smooth 6 96 (1995), 115–147. MR1816214 cubic surface. We define the following candidate for apa- [Bea77] A. Beauville, Variétés de Prym et jacobiennes in- rameter space of 풜6: termédiaires, Ann. Sci. Ecole´ Norm. Sup. (4) 10 (1977), 27∶1 309–391. MR0472843 ℋ = {푓 ∶ 퐶 ⟶ ℙ1 ∣ 푀표푛(푓) = 푊(퐸 ), 푓 is 퐸6 6 [BL04] Ch. Birkenhake and H. Lange, Complex abelian va- ramified over 24 points with ramification rieties, 2nd ed., Grundlehren der Mathematischen Wis- pattern determined by root reflexion}. senschaften, vol. 302, Springer Verlag, Berlin, 2004. [Deb89] O. Debarre, Sur le problème de Torelli pour les variétés This is a special Hurwitz scheme that incorporates the in- de Prym, Amer. J. Math. 111 (1989), 111–134. MR0980302 cidence correspondence of lines on a cubic surface in the [Don81] R. Donagi, The tetragonal construction, Bull. Amer. 푓 1 Math. Soc. (N.S.) 4 (1981), 181–185. MR0001219 monodromy of the covering; that is, for each [퐶 ⟶ ℙ ] ∈ [Don92] R. Donagi, The fibers of the Prym map, Contemp.

ℋ퐸6 we have a correspondence on 퐶 of degree 10 that be- Math. 136 (1992), 55–125. MR1188194 haves just like in the family of lines constructed above. If 훾 [FS82] R. Friedman and R. Smith, The generic Torelli theorem for the Prym map, Invent. Math. 67 (1982), 473–490. 5 푁 99K That is, there is a dominant rational map ℙ 풜푔. MR0664116

1322 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 9 [Kan82] V.I. Kanev, A global Torelli theorem for Prym varieties at a general point, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 244–268. MR0651647 [LO11] H. Lange and A. Ortega, Prym varieties of cyclic coverings, Geom. Dedicata 150 (2011), 391–403. MR2753712 [LO18] H. Lange and A. Ortega, On the Prym map of cyclic coverings, Arch. Math. 111 (2018), 621–631. MR3877472 [MN14] V. Marcucci and J.C. Naranjo, Prym varieties of double coverings of elliptic curves, Int. Math. Res. Notices 6 (2014), 1689–1698. MR3180606 [MP12] V. Marcucci and G.P. Pirola, Generic Torelli for Prym varieties of ramified coverings, Compos. Math. 148 (2012), 1147–1170. MR2956039 ual [Mum83] D. Mumford, On the Kodaira dimension of the Siegel r ann modular variety, Lecture Notes in Math., vol. 997, Springer, Ou Berlin, 1983. MR0688651 n [Kan89] V.I. Kanev, Spectral curves, simple lie algebras, and ratio Prym-Tjurin varieties, Vol. 49, Amer. Math. Soc., Providence, leb members! RI, 1989. MR1013158 ce f AMS [Mum74] D. Mumford, Prym varieties I, Contributions to o Analysis (a collection of papers dedicated to Lipman Bers), 1974, pp. 325–350. MR0379510 [NO19] J.C. Naranjo and A. Ortega, Generic injectivity of the Prym map for double ramified coverings. with an appendix by Join us as we honor our AMS members Alessandro Verra, Trans. Amer. Math. Soc. 371 (2019), 3627– via “AMS Day,” a day of specials on AMS 3646. MR3896124 publications, membership, and much [NR95] D.S. Nagaraj and S. Ramanan, Polarisations of type more! Stay tuned on social media and (1, 2, … , 2) on abelian varieties, Duke Math. J. 80 (1995), membership communications for 157–194. MR1360615 details about this exciting day. [Wel87] G. Welters, Recovering the curve data from a general Prym variety, Amer. J. Math. 109 (1987), 165–182. Spread the word about #AMSDay today! MR0878204 [Wir95] W. Wirtinger, Untersuchen über Thetafunktionen, Teub- ner, Leipzig, 1895.

Angela Ortega Credits onday, Opening image is courtesy of Oliver Labs (MO-Labs.com). M Figures are courtesy of Angela Ortega. ovember 30th Photo of Angela Ortega is courtesy of Alma Ortega. N

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