DOCSLIB.ORG

The Prym Map: from Coverings to Abelian Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 dif- ferential 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 vari- eties and their moduli spaces. Coverings of curves and Prym varieties. Consider a finite covering 푓 ∶ 퐶˜ → 퐶 of degree 푑 between two smooth pro- jective 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.
Load more

Recommended publications
  • Vector Bundles and Brill–Noether Theory
    MSRI Series Volume 28, 1995 Vector Bundles and Brill{Noether Theory SHIGERU MUKAI Abstract. After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose sev- eral problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical determinant with five linearly independent global sections on a non-tetragonal curve of genus 7 is a smooth Fano threefold of genus 7. As a natural generalization of line bundles, vector bundles have two important roles in algebraic geometry. One is the moduli space. The moduli of vector bun- dles gives connections among different types of varieties, and sometimes yields new varieties that are difficult to describe by other means. The other is the linear system. In the same way as the classical construction of a map to a pro- jective space, a vector bundle gives rise to a rational map to a Grassmannian if it is generically generated by its global sections. In this article, we shall de- scribe some results for which vector bundles play such roles. They are obtained from an attempt to generalize Brill–Noether theory of special divisors, reviewed in Section 2, to vector bundles. Our main subject is rank-two vector bundles with canonical determinant on a curve C with as many global sections as pos- sible: especially their moduli and the Grassmannian embeddings of C by them (Section 4).
    [Show full text]
  • Arxiv:Math/9902145V1 [Math.AG] 25 Feb 1999 by Oprto Ntaie(Otatn.C1-T303) Ohau Both CI1*-CT93-0031)
    To appear in Journal f¨ur die reine und angewandte Mathematik RESTRICTION OF THE POINCARE´ BUNDLE TO A CALABI-YAU HYPERSURFACE INDRANIL BISWAS AND L. BRAMBILA-PAZ 1. Introduction Let X be a compact connected Riemann surface of genus g, where g ≥ 3. Denote by Mξ := M(n, ξ) the moduli space of stable vector bundles over X of rank n and fixed determinant ξ. If the degree deg(ξ) and the rank n are coprime, then there is a universal family of vector bundles, U, over X parametrized by Mξ. This family is unique up to tensoring by a line bundle that comes from Mξ. We fix one universal family over X ×Mξ and call it the Poincar´ebundle. For any x ∈ X, let Ux denote the vector bundle over Mξ obtained by restricting U to x ×Mξ. It is known that U (see [BBN]) and Ux (see [NR] and [Ty]) are stable vector bundles with respect to any polarization on X ×Mξ and Mξ respectively. A smooth anti-canonical divisor D on Mξ is an example of a Calabi-Yau variety, i.e., it is connected and simply connected with trivial canonical line bundle. The Calabi-Yau varieties are of interest both in string theory and in algebraic geometry. In this paper we consider the restrictions of U and Ux to X×D and x×D respectively, where x ∈ X and D is a smooth anti-canonical divisor. Denote such restrictions by UD and (UD)x respectively. In Theorem 2.5 and Corollary 2.6 we prove the following : If n ≥ 3, then the vector bundle (UD)x is stable with respect to any polarization on D.
    [Show full text]
  • Prym Covers, Theta Functions and Kobayashi Curves in Hilbert
    Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces Martin M¨oller November 1, 2018 Abstract Algebraic curves in Hilbert modular surfaces that are totally geodesic for the Kobayashi metric have very interesting geometric and arithmetic properties, e.g. they are rigid. There are very few methods known to construct such algebraic geodesics that we call Kobayashi curves. We give an explicit way of constructing Kobayashi curves using deter- minants of derivatives of theta functions. This construction also allows to calculate the Euler characteristics of the Teichm¨uller curves constructed by McMullen using Prym covers. Contents 1 Background.............................. 4 2 Prymvarieties ............................ 10 3 A Prym variety with (1, 2)-polarization. 15 X S 4 Euler characteristic of WD and WD ................ 19 5 An invariant and possible generalizations . 23 Introduction We call Kobayashi curves algebraic curves on Hilbert modular surfaces that are arXiv:1111.2624v1 [math.AG] 10 Nov 2011 totally geodesic for the Kobayashi metric. They are rigid and interesting both from geometric and arithmetic point of view and there are very few methods to construct Kobayashi curves. This paper can be read from two perspectives, the construction of Kobayashi curves and the calculation of invariants of Teichm¨uller curves. The most obvious Kobayashi curve on a Hilbert modular surface XD = 2 o o 2 H /SL( D D∨ ) is the image of the diagonal in H . One can twist it by a ⊕ + σ matrix M GL2 (Q(√D), i.e. consider the image of z (Mz,M z) and obtain further Kobayashi∈ curves, also known as Hirzebruch-Zagier7→ cycles or Shimura curves.
    [Show full text]
  • Numerical Computation of Endomorphism Rings of Jacobians Nils Bruin, Jeroen Sijsling, and Alexandre Zotine
    THE OPEN BOOK SERIES 2 ANTS XIII Proceedings of the Thirteenth Algorithmic Number Theory Symposium Numerical computation of endomorphism rings of Jacobians Nils Bruin, Jeroen Sijsling, and Alexandre Zotine msp THE OPEN BOOK SERIES 2 (2019) Thirteenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2019.2.155 Numerical computation of endomorphism rings of Jacobians Nils Bruin, Jeroen Sijsling, and Alexandre Zotine We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties. 1. Introduction Let k be a field of characteristic 0 that is finitely generated over Q. We choose an embedding of k into C. In this article, we consider nonsingular, complete, absolutely irreducible algebraic curves C over k of genus g. We represent such a curve C by a possibly singular affine plane model Ce V f .x; y/ D 0; where f .x; y/ 2 kTx; yU: (1-1) Associated to C is the Jacobian variety J D Jac.C/ representing Pic0.C/. Classical results by Abel and Jacobi establish ∼ 0 1 ∗ ∼ g 2g J.C/ D H .CC; C / =H1.C.C/; Z/ D C =Z ; for a suitable g × 2g matrix , called a period matrix of C. Let J1 D Jac.C1/ and J2 D Jac.C2/ be two such Jacobian varieties.
    [Show full text]
  • Arxiv:2007.09646V1 [Math.AG]
    SHIMURA CURVES IN THE PRYM LOCI OF RAMIFIED DOUBLE COVERS PAOLA FREDIANI AND GIAN PAOLO GROSSELLI Aδ Abstract. We study Shimura curves of PEL type in the space of polarised abelian varieties p gener- ically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is P1. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci. 1. Introduction Let Rg,b be the moduli space parametrising isomorphism classes of triples [(C,η,B)] where C is a smooth complex projective curve of genus g, B is a reduced effective divisor of degree b on C and η is a 2 line bundle on C such that η = OC (B). To such data it is associated a double cover of C, f : C˜ → C branched on B. The Prym variety associated to [(C,η,B)] is the connected component containing the origin of the kernel of the norm map Nmf : JC˜ → JC. For b > 0, ker Nmf is connected. It is a polarised abelian b ˜ variety of dimension g − 1+ 2 , denoted by P (C,η,B) or equivalently P (C, C). Let Ξ be the restriction on P (C,˜ C) of the principal polarisation on JC˜. For b> 0 the polarisation Ξ is of type δ = (1,..., 1, 2,..., 2).
    [Show full text]
  • Riemann Surfaces of Genus $1+ Q^ 2$ with $3 Q^ 2$ Automorphisms
    RIEMANN SURFACES OF GENUS 1+ q2 WITH 3q2 AUTOMORPHISMS ANGEL CAROCCA AND SEBASTIAN´ REYES-CAROCCA Abstract. In this article we classify compact Riemann surfaces of genus1+ q2 with a group of automorphisms of order 3q2, where q is a prime number. We also study decompositions of the corresponding Jacobian varieties. 1. Introduction and statement of the results The study and classification of groups of automorphisms of compact Riemann sur- faces (or complex projective algebraic curves) is a classical problem which has attracted considerable interest since a long time. Regarding this issue, in the late nineteenth cen- tury a fundamental result was obtained by Hurwitz, who succeeded in proving that the full automorphism group of a compact Riemann surface of genus g > 2 is finite, and that its order is at most 84g − 84. Later, this problem acquired a new relevance when its relationship with Teichm¨uller and moduli spaces was developed. An interesting problem is to study and describe those compact Riemann surfaces whose automorphism groups share a common property. The most prominent example concerning that are Hurwitz curves; namely, those Riemann surfaces possessing the maximal possible number of automorphisms. Nowadays, it is classically known that Hurwitz curves correspond to regular covers of the projective line ramified over three values, marked with 2, 3 and 7. Another well-known example is the cyclic case, which was considered, among others, by Wiman. In [41], he showed that the largest cyclic group of automorphisms of a Riemann surface of genus g > 2 has order at most 4g + 2. arXiv:1911.04310v2 [math.AG] 22 Jul 2020 Furthermore, the so-called Wiman curve of type I given by y2 = x2g+1 − 1 shows that this upper bound is attained for each g; see [15].
    [Show full text]
  • PRYM VARIETIES and PRYM MAP MINI-COURSE 1. Basics on Abelian
    PRYM VARIETIES AND PRYM MAP MINI-COURSE ANGELA ORTEGA 1. Basics on abelian varieties Through this notes we work on C. Definition 1.1. A complex torus A is a quotient V=Λ, with V ' Cg a C- vector space and Λ ' Z2g a full rang lattice inside V . A polarization on A is an ample line bundle1 L on A. An abelian variety is a complex torus admitting a polarization, so (A; L) is polaraized abelian variety. Remark 1.2. In particular, with the addition operation inherited from V , an abelian variety is an abelian group. By definition of ampleness, given a line bundle L on A we have that the map 0 ⊗k ∗ 'L⊗k : A,! PH (A; L ) x 7! [s0(x): s1(x): ··· : sN (x)]; defined by the sections of L⊗k is an embedding for some k > 1. In fact, in the case of polarized abelian varieties it suffices to take k = 3. Then an abelian variety is also a projective variety. Different encarnations of a polarization on A. The following data are equivalent: 2 • A first Chern class c1(L) 2 H (A; ZZ) of an ample line bundle L on A. • A non degenerated alternating form E : V × V ! R such that E(Λ; Λ) ⊂ Z and E(iv; iw) = E(v; w). • A non degenerated Hermitian form H : V × V ! C with H(Λ; Λ) ⊂ Z. 0 • An isogeny φL : A ! Ab := Pic (A) ∗ • A Weil divisor Θ ⊂ A such that the subgroup fx 2 A j txΘ ∼ Θg is finite. Let E be an alternating form representing a polarization on A = V=Λ.
    [Show full text]
  • Hyperelliptic Prym Varieties and Integrable Systems
    Commun. Math. Phys. 221, 169 – 196 (2001) Communications in Mathematical Physics © Springer-Verlag 2001 Hyperelliptic Prym Varieties and Integrable Systems Rui Loja Fernandes1,, Pol Vanhaecke2 1 Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal. E-mail: [email protected] 2 Université de Poitiers, Département de Mathématiques, Téléport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil Cedex, France. E-mail: [email protected] Received: 12 December 2000 / Accepted: 26 March 2001 Abstract: We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice a˙i = ai(ai−1 − ai+1), i = 1,...,n, an+1 = a1, is an affine part of a hyperelliptic Prym variety, obtained by removing n translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable. Contents 1. Introduction ................................. 169 2. Hyperelliptic Prym Varieties ......................... 171 3. The Hyperelliptic Prym Systems ...................... 175 4. The Periodic Toda Lattices and KM Systems ................ 182 5. Painlevé Analysis .............................. 188 6. Example: n = 5 ............................... 192 1. Introduction In this paper we introduce two algebraic completely integrable (a.c.i.) systems, similar to the even and odd Mumford systems (see [12] for the odd systems and [15] for the even systems). By a.c.i. we mean that the general level set of the momentum map is Supported in part by FCT-Portugal through the Research Units Pluriannual Funding Program, European Research Training Network HPRN-CT-2000-00101 and grant POCTI/1999/MAT/33081.
    [Show full text]
  • The Norm Map on the Compactified Jacobian, the Prym Stack Spectral
    Dottorato di Ricerca in Matematica XXXII Ciclo Dottorato in Matematica The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs RAFFAELE CARBONE Nome e Cognome del dottorando Firma arXiv:2006.13034v1 [math.AG] 22 Jun 2020 FILIPPO VIVIANI Docente Guida/Tutor: Prof. Firma ANGELO LOPEZ Coordinatore: Prof. Firma Contents Introduction and statement of results 3 0.1. Preliminaries 6 0.2. Direct image of generalized divisors and Norm map 10 0.3. Spectral data for G-Higgs pairs 13 Chapter 1. Preliminaries 21 1.1. Rank and degree of coherent sheaves 21 1.2. Moduli spaces of torsion-free sheaves “of rank 1” 26 1.3. Moduli spaces of Higgs pairs 31 1.4. The spectral correspondence for Higgs pairs 34 Chapter 2. The Norm map on the compactified Jacobian 43 2.1. Review of the Norm map 44 2.2. Review of generalized divisors 49 2.3. The direct and inverse image for generalized divisors and generalized line bundles 56 2.4. The direct and inverse image for families of generalized divisors 76 2.5. The Norm and the inverse image for families of torsion-free rank-1 sheaves 78 Chapter 3. The fibers of the Norm map and the Prym stack 85 3.1. The fibers of the Hilbert-Chow morphism 86 3.2. The fibers of the direct image between Hilbert schemes 87 3.3. The fibers of the Norm map and Pr(X; Y ) 90 Chapter 4. Spectral data for G-Higgs pairs 93 4.1. SL(r; C)-Higgs pairs 93 4.2.
    [Show full text]
  • A Prym Variety with Everywhere Good Reduction Over Q( √
    A PRYM VARIETY WITH EVERYWHEREp GOOD REDUCTION OVER Q( 61) NICOLAS MASCOT, JEROEN SIJSLING, AND JOHN VOIGHT Abstract. We compute an equation for a modularp abelian surface A that has everywhere good reduction over the quadratic field K = Q( 61) and that does not admit a principal polarization over K. 1. Introduction A foundational theorem of Abrashkin [1] and Fontaine [14] asserts that there is no abelian variety over Q with everywhere good reduction; this theorem is an analogue in arithmetic geometry of Minkowski's theorem that every number field has a ramified prime. More gener- ally, the finiteness of the set of isomorphism classes of abelian varieties over a given number field of given dimension and with good reduction outside a given finite set is a key ingredient in the proof of Faltings [12] that a curve of genus at least two defined over a number field has finitely many rational points. For this reason and many others, it remains an important task to explicitly study abelian varieties over number fields with controlled ramification [3, 29]. Moving beyond elliptic curves, Demb´el´e{Kumar[10] have given examples of abelian sur- faces with everywhere good reduction over real quadratic fields. More recently, Demb´el´ep [9] gave a classification of abelian varieties with everywhere good reduction over Q( d) with d = 53; 61; 73: assuming the generalized Riemann hypothesis (GRH), he proves that every such abelian variety is isogenous to the power of a unique abelian surface Ad. Moreover, for d = 53; 73 he was able to find an explicit equation for a genus 2 curve whose Jacobian is isogenous to Ad.
    [Show full text]
  • Prym Varieties and Teichmüller Curves
    Prym varieties and Teichm¨uller curves Curtis T. McMullen∗ 15 April, 2005 Abstract This paper gives a uniform construction of infinitely many primitive Teichm¨uller curves V g for g =2, 3 and 4. ⊂M Contents 1 Introduction............................ 1 2 Teichm¨ullercurves ........................ 5 3 Prymvarieties .......................... 8 4 Curvesystems .......................... 13 5 Examplesingenus2,3and4 .................. 15 6 Quadraticdifferentials . 19 1 Introduction A Teichm¨uller curve V is a totally geodesic algebraic curve in the ⊂ Mg moduli space of Riemann surfaces of genus g. We say V is primitive if it is generated by an Abelian differential of min- imal genus. Every Teichm¨uller curve has a unique primitive representative in its commensurability class (see 2). § In this paper we will show: Theorem 1.1 There exist infinitely many primitive Teichm¨uller curves V ⊂ for g = 2, 3 and 4. Mg The case g = 2 was treated previously in [Mc1] using Jacobians with real multiplication. In this paper we use Prym varieties and pseudo-Anosov diffeomorphisms to present a uniform construction for genus 2 g 4. ≤ ≤ We also construct new closed, SL2(R)-invariant loci in Ω for g 5, Mg ≤ and new Teichm¨uller curves generated by strictly quadratic differentials. ∗Research supported in part by the NSF. 2000 Mathematics Subject Classification: Primary 32G15, Secondary 14H40, 37D50, 57M. 1 Prym varieties. Let Ω(X) denote the vector space of holomorphic 1-forms on X , and let ρ : X X be a holomorphic involution. The variety ∈ Mg → − ∗ − Prym(X, ρ) = (Ω(X) ) /H1(X, Z) is the subtorus of the Jacobian of X determined by the negative eigenspace of ρ.
    [Show full text]
  • A Singular Symplectic Variety of Dimension 6 with a Lagrangian Prym Fibration Tommaso Matteini
    A singular symplectic variety of dimension 6 with a Lagrangian Prym fibration Tommaso Matteini To cite this version: Tommaso Matteini. A singular symplectic variety of dimension 6 with a Lagrangian Prym fibration. manuscripta mathematica, Springer Verlag, 2016, 149 (1), 10.1007/s00229-015-0777-z. hal-01288397 HAL Id: hal-01288397 https://hal.archives-ouvertes.fr/hal-01288397 Submitted on 15 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A singular symplectic variety of dimension 6 with a Lagrangian Prym fibration Tommaso Matteini IRMA - Université de Strasbourg 7 rue René Descartes, 67000 Strasbourg - France Telephone: +33 (0)3 68 85 01 53 E-mail address: [email protected] Abstract A projective symplectic variety P of dimension 6, with only finite (2,0) quotient singularities, π(P)=0 and h (Psmooth)=1, is described as a relative compactified Prym variety of a family of genus 4 curves with involution. It is a Lagrangian fibration associated to a K3 surface double cover of a generic cubic surface. It has no symplectic desingu- larization. MSC codes: 14D06, 14J40, 14B05. arXiv:1403.5523v3 [math.AG] 17 Jul 2015 1 Introduction In this paper, we work in the setting of complex projective varieties.
    [Show full text]