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Polarizations of Prym varieties for Weyl groups via abelianization

Herbert Lange, Christian Pauly (2009)

Journal of the European Mathematical Society

Let π : Z → 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 λ consider the curve Y = Z / Stab ( λ ) . The Kanev correspondence defines an abelian subvariety P λ of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of Jac ( Y ) to P λ 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...

Singularities of 2 Θ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

We consider the linear system | 2 Θ 0 | of second order theta functions over the Jacobian J C of a non-hyperelliptic curve C . A result by J.Fay says that a divisor D ∈ | 2 Θ 0 | contains the origin 𝒪 ∈ J C with multiplicity 4 if and only if D contains the surface C - C = { 𝒪 ( p - q ) ∣ p , q ∈ C } ⊂ J C . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing 𝒪 with multiplicity 6 , divisors containing the fourfold C 2 - C 2 = { 𝒪 ( p + q - r - s ) ∣ p , q , r , s ∈ C } , and divisors singular along C - C , using the third exterior...

Singularities of theta divisors and the geometry of 𝒜 5

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

We study the codimension two locus H in 𝒜 g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class [ H ] ∈ C H 2 ( 𝒜 g ) for every g . For g = 4 , this turns out to be the locus of Jacobians with a vanishing theta-null. For g = 5 , via the Prym map we show that H ⊂ 𝒜 5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of 𝒜 5 ¯ and show that the component N 0 ' ¯ of the Andreotti-Mayer...

Sommes de Dedekind elliptiques et formes de Jacobi

Abdelmejid Bayad (2001)

Annales de l’institut Fourier

À partir des formes de Jacobi D L ( z , ϕ ) , on construit une somme de Dedekind elliptique. On obtient ainsi un analogue elliptique aux sommes multiples de Dedekind construites à partir des fonctions cotangentes, étudiées par D. Zagier. En outre, on établit une loi de réciprocité satisfaite par ces nouvelles sommes. Par une procédure de limite, on peut retrouver la loi de réciprocité remplie par les sommes multiples de Dedekind classiques. D’autre part, en les spécialisant en des paramètres de points de 2- division,...

Currently displaying 81 – 100 of 179