Affiliation:
1. Dipartimento di Matematica , Università di Pavia , Via Ferrata, 5, 27100 , Pavia , Italy
Abstract
Abstract
Let
Jg
be the Jacobian locus and let
P
g+1
be the Prym locus, in the moduli space Ag
of principally polarized abelian varieties of dimension g, for g ≥ 7. We study the extrinsic geometry of
Jg
⊂
P
g+1
, under the inclusion provided by the theory of generalized Prym varieties introduced by Beauville. More precisely, we address the problem if certain geodesic curves, defined with respect to the Siegel metric of Ag
, starting at a Jacobian variety [JC] ∈ Ag
of a curve [C] ∈ Mg
and with direction ζ ∈ T[JC]
Jg
, are locally contained in
P
g+1
. The result is that for a general JC, the answer is negative, if the rank k of ζ is such that k < Cliff C − 3, where Cliff C denotes the Clifford index of C.
Reference21 articles.
1. E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I. Springer 1985. MR770932 Zbl 0559.14017
2. A. Beauville, Prym varieties and the Schottky problem. Invent. Math. 41 (1977), 149–196. MR572974 Zbl 0333.14013
3. A. Beauville, Complex algebraic surfaces. Cambridge Univ. Press 1983. MR732439 Zbl 0512.14020
4. L. Caporaso, Linear series on semistable curves. Int. Math. Res. Not. volume 2011, issue 13, (2011), 2921–2969. MR2817683 Zbl 1231.14007
5. F. Catanese, M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures. In: Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata’s sixtieth birthday, volume 74 of Adv. Stud. Pure Math., 73–102, Math. Soc. Japan, Tokyo 2017. MR3791209 Zbl 1388.14037