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Summands of theta divisors on Jacobians
Part of
Thomas Krämer
Journal:

Compositio Mathematica / Volume 156 / Issue 7 / July 2020

Published online by Cambridge University Press:

08 July 2020, pp. 1457-1475

Print publication:

July 2020
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We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

14 - The Calogero-Franc¸oise Integrable System: Algebraic Geometry, Higgs Fields, and the Inverse Problem
- By Steven Rayan, Thomas Stanley, Jacek Szmigielski
Edited by Ron Donagi, University of Pennsylvania, Tony Shaska, Oakland University, Michigan
Book:

Integrable Systems and Algebraic Geometry

Published online:

19 March 2020

Print publication:

02 April 2020, pp 356-382
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Summary

We review the Calogero–Françoise integrable system, which is a generalization of the Camassa–Holm system. We express solutions as (twisted) Higgs bundles, in the sense of Hitchin, over the projective line. We use this point of view to (a) establish a general answer to the question of linearization of isospectral flow and (b) demonstrate, in the case of two particles, the dynamical meaning of the theta divisor of the spectral curve in terms of mechanical collisions. Lastly, we outline the solution to the inverse problem for CF flows using Stieltjes’ continued fractions.

Revêtements des courbes en caractéristique p>0 et ordinarité
Michel Raynaud
Journal:

Compositio Mathematica / Volume 123 / Issue 1 / August 2000

Published online by Cambridge University Press:

04 December 2007, pp. 73-88

Print publication:

August 2000
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Let X be a proper, smooth, connected curve, defined over an algebraically closed field of characteristic p>0 and of genus g ≥ 2. We show that there exists a finite solvable group G, of order prime to p, and a Galois étale cover Y → X, with Galois group G, which is not ordinary.

Divisors on Principally Polarized Abelian Varieties
CHRISTOPHER D. HACON
Journal:

Compositio Mathematica / Volume 119 / Issue 3 / December 1999

Published online by Cambridge University Press:

04 December 2007, pp. 321-329

Print publication:

December 1999
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The purpose of this paper is to show how generalizations of generic vanishing theorems to a${\mirrored Q}$-divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).

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