Prym-Tyurin Varieties Research Articles

A pointed Prym–Petri Theorem

We construct pointed Prym–Brill–Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym–Petri map and prove a pointed version of the Prym–Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters [Ann. Sci. Ëcole Norm. Sup. (4) 18 (1985), pp. 671–683] and De Concini–Pragacz [Math. Ann. 302 (1995), pp. 687–697] on the unpointed case. Finally, we show that Prym varieties are Prym–Tyurin varieties for Prym–Brill–Noether curves of exponent enumerating standard shifted tableaux times a factor of 2 2 , extending to the Prym setting work of Ortega [Math. Ann. 356 (2013), pp. 809–817].

The Brill–Noether curve and Prym-Tyurin varieties

We prove that the Jacobian of a general curve C of genus g=2a+1, with g>4, can be realized as a Prym-Tyurin variety for the Brill-Noether curve W^1_{a+2}(C). As a consequence of this result we are able to compute the class of the sum of the secant divisors for the curve C, embedded with a complete linear series g^{a-1}_{3a-2}

Prym–Tyurin varieties using self-products of groups

Given Prym–Tyurin varieties of exponent q with respect to a finite group G, a subgroup H and a set of rational irreducible representations of G satisfying some additional properties, we construct a Prym–Tyurin variety of exponent [ G : H ] q in a natural way. We study an example of this result, starting from the dihedral group D p for any odd prime p. This generalizes the construction of [H. Lange, S. Recillas, A.M. Rojas, A family of Prym–Tyurin varieties of exponent 3, J. Algebra 289 (2005) 594–613] for p = 3 . Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example.

Polarizations of Prym varieties for Weyl groups via abelianization

Let \pi: Z \to X be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group G . For any dominant weight \lambda consider the curve Y = Z/\operatorname{Stab}(\lambda) . The Kanev correspondence defines an abelian subvariety P_\lambda of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of \operatorname{Jac}(Y) to P_\lambda in some cases. In particular, in the case of the group E_8 we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi–Prym variety to the moduli stack of principal G -bundles on the curve X .

Prym-Tyurin varieties via Hecke algebras

Let G denote a finite group and π : Z → Y a Galois covering of smooth projective curves with Galois group G. For every subgroup H of G there is a canonical action of the corresponding Hecke algebra ℚ[H\G/H] on the Jacobian of the curve X = Z/H. To each rational irreducible representation 𝒲 of G we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve X and thus an abelian subvariety P of the Jacobian JX. We give sufficient conditions on 𝒲, H, and the action of G on Z for P to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.

Correspondences with split polynomial equations

We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tyurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.

Products of Jacobians as Prym-Tyurin varieties

Let X 1, ..., X m denote smooth projective curves of genus g i ≥ 2 over an algebraically closed field of characteristic 0 and let n denote any integer at least equal to \({1+\max_{i=1}^m g_i}\) . We show that the product JX 1 × ... × JX m of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent n m-1. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.

A Galois-theoretic approach to Kanev’s correspondence

Let G be a finite group, \(\Lambda\) an absolutely irreducible \({\mathbb{Z}}[G]\) -module and w a weight of \(\Lambda\) . To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym–Tyurin varieties.

Prym varieties associated to graphs

We present a Prym construction which associates abelian varieties to vertex-transitive strongly regular graphs. As an application we construct Prym–Tyurin varieties of arbitrary exponent ⩾3, generalizing a result by Lange, Recillas and Rojas.

Prym–Tyurin varieties coming from correspondences with fixed points

Our main theorem is an improvement of the criterion of Kanev about Prym–Tyurin varieties induced by correspondences, which includes correspondences with fixed points. We give some examples of Prym–Tyurin varieties using this criterion.

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f i :X → X i , i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J X , the Prym variety P(f 1, f 2) of the pair (f 1, f 2). In some cases we can compute the type of the restriction of the canonical principal polarization of J X . We obtain 2 families of Prym-Tyurin varieties of exponent 6.

A family of Prym–Tyurin varieties of exponent 3

We investigate a family of correspondences associated to étale coverings of degree 3 of hyperelliptic curves. They lead to Prym–Tyurin varieties of exponent 3. We identify these varieties and derive some consequences.