On the Jacobian locus in the Prym locus and geodesics
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Sara Torelli
Abstract
Let Jg be the Jacobian locus and let Pg+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 ⊂ Pg+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 Pg+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.
Acknowledgements
I am very grateful to Gian Pietro Pirola, for the interesting and profound research conversations that we have had over time and that have stimulated this work. I would also like to thank Juan Carlos Naranjo and Alessandro Ghigi, for their helpful explanations and comments.
Funding: The author was supported by PRIN 2017 Moduli spaces and Lie Theory, INdAM - GNSAGA, FAR 2016 (Pavia) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici” and MIUR, Programma Dipartimenti di Eccellenza (2018−2022) - Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia.
Communicated by: R. Cavalieri
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Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
Articles in the same Issue
- Frontmatter
- The Malgrange–Galois groupoid of the Painlevé VI equation with parameters
- Explicit Nikulin configurations on Kummer surfaces
- Irregular surfaces on hypersurfaces of degree 4 with non-degenerate isolated singularities
- Counting isolated points outside the image of a polynomial map
- An inverse approach to hyperspheres of prescribed mean curvature in Euclidean space
- Bridgeland stability conditions on surfaces with curves of negative self-intersection
- Computation of Dressians by dimensional reduction
- A few more extensions of Putinar’s Positivstellensatz to non-compact sets
- On the Jacobian locus in the Prym locus and geodesics
- Weierstrass semigroups for maximal curves realizable as Harbater–Katz–Gabber covers
