Prym varieties

Prym varieties
Name	Prym varieties
Field	Algebraic geometry
Introduced	19th century

Prym varieties are principally polarized complex abelian varieties arising as complementary abelian subvarieties associated to finite morphisms of algebraic curves. They occupy a central role in the theory of Jacobians of curves, link classical algebraic geometry with modern Hodge theory, and appear in problems relating to integrable systems, Schottky problem, and the geometry of moduli spaces such as Moduli space of curves.

Introduction

The construction of Prym varieties originated in classical work on double covers of compact Riemann surfaces and later was systematized in the algebraic framework by geometers studying Abelian varieties, Theta functions, and the geometry of Algebraic curves. Historically connected figures include Friedrich Prym, Riemann, Igor Shafarevich, and David Mumford. Prym varieties provide refined invariants beyond the Jacobian variety: given a finite morphism of curves with involution, one extracts an abelian subvariety (or quotient) capturing the anti-invariant part of the Jacobian under the involution, which interacts with canonical polarizations and with maps to moduli such as Siegel modular variety.

Construction and Definitions

Let f: C' → C be a finite morphism of smooth projective curves over the complex numbers, frequently specialized to an unramified or étale double cover with associated involution σ on C'. The natural map f* : Jac(C) → Jac(C') and the norm map Nm_f : Jac(C') → Jac(C) induce decompositions of Jacobians under pullback and pushforward. The Prym variety is defined as the connected component of ker(Nm_f) containing the origin, equivalently the identity component of the anti-invariant subspace (1−σ)Jac(C') when σ generates a Galois group of order two. Key contributors to formal definitions include Andreotti, Mumford, and Matsusaka in the context of polarizations and abelian subvarieties. Generalizations treat cyclic covers with Galois group a finite abelian group and relate to representations of groups studied by Noether and Artin.

Basic Properties and Polarization

A Prym variety P(f) is an abelian variety of dimension g(C') − g(C), where g denotes genus; in the classical étale double cover case this dimension equals g(C) − 1. The induced polarization on P(f) is typically of type (1, ...,1,2,...,2) and need not be principal except in special situations such as \'etale double covers of curves of genus g≥2 where P(f) often admits a principal polarization; this phenomenon is central to the study of the Torelli theorem and its Prym analogues. Important results include the description of the Rosati involution coming from the canonical principal polarization of Jac(C') and compatibility with the norm and trace maps studied by Poincaré and Weil. Monodromy of families of Prym varieties connects to representations of mapping class groups studied by Nielsen and Dehn and to local systems considered by Deligne.

Examples and Classical Cases

Classical examples arise from unramified double covers of hyperelliptic curves and from étale double covers branched at four points leading to Prym varieties isomorphic to Jacobians of other curves, a phenomenon explored by Schottky and Jung. The case of étale double covers of a genus 3 curve yields principally polarized Prym surfaces often isomorphic to the Jacobian of a genus 2 curve, linking to the theory of Kummer surfaces and to explicit constructions by Coble and Hudson. Another celebrated example is the construction of Prym varieties via cyclic triple covers related to the works of Wiman and Hurwitz on automorphisms of curves. Connections with K3 surfaces occur where Nikulin involutions produce Prym-type decompositions in the Picard lattice studied by Nikulin.

Moduli and Torelli-type Theorems

The Prym map sends a cover f: C' → C to the isomorphism class of the polarized abelian variety P(f), defining a morphism from a Hurwitz-type moduli space of covers to a suitable subspace of the Siegel modular variety. Fundamental questions ask when this Prym map is injective, generically finite, or dominant; significant theorems include the generic injectivity results for low genera obtained by Mumford, Beauville, and Debarre. Prym–Torelli theorems assert reconstruction of the cover (or of the pair (C', C)) from the polarized Prym in many cases, paralleling the classical Torelli theorem for Jacobians due to Torelli and refined by Andreotti and Liu. Degeneration techniques use the Deligne–Mumford compactification and relate to boundary strata studied by Witten and Harris.

Applications and Connections

Prym varieties appear in the study of integrable systems such as the Kadomtsev–Petviashvili equation via algebro-geometric solutions associated to linear flows on abelian varieties, a perspective developed by Novikov, Dubrovin, and Krichever. They play roles in questions about rationality and unirationality of moduli spaces examined by Harris, Morrison, and Farkas. In number theory, Prym varieties contribute to the study of rational points on curves and to explicit descent techniques in the manner of Selmer and Cassels. In mathematical physics, Prym varieties enter string dualities where compactifications involve Calabi–Yau manifolds and their intermediate Jacobians explored by Clemens and Griffiths.

Computational Aspects and Explicit Cases

Explicit computations of Prym varieties exploit period matrices, theta functions, and algebraic correspondences; computational work uses techniques from Mumford's theory of theta groups and modern algorithmic approaches by researchers influenced by SageMath development and computational algebraic geometry initiatives at Institut des Hautes Études Scientifiques and Max Planck Institute for Mathematics. Examples computed explicitly include Prym varieties of double covers with small branching loci, equations for associated Kummer varieties, and endomorphism rings in cases with extra automorphisms studied by Faltings and Raynaud. Effective algorithms for computing the polarization type, period matrix, and principally polarized isogeny classes leverage point-counting methods used in arithmetic geometry by Weil and computational packages inspired by projects at University of Cambridge and Princeton University.

Category:Abelian varieties