Algebraic geometry
A density result for real hyperelliptic curves
Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1219-1224.

Let {+,} be the two points above ∞ on the real hyperelliptic curve H:y2=(x21)i=12g(xai). We show that the divisor ([+][]) is torsion in Jac J for a dense set of (a1,a2,,a2g)(1,1)2g. In fact, we prove by degeneration to a nodal P1 that an associated period map has derivative generically of full rank.

Soient {+,} les deux points de la courbe hyperelliptique réelle H:y2=(x21)i=12g(xai) au-dessus du point ∞ de P1. On montre que le diviseur ([+][]) est de torsion dans Jac J pour un ensemble dense de (a1,a2,,a2g)(1,1)2g. En fait, on démontre par réduction à un P1 avec des points doubles que la dérivée d'un morphisme de périodes est génériquement surjectif.

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Published online:
DOI: 10.1016/j.crma.2016.10.014
Lawrence, Brian 1

1 Department of Mathematics, Building 380, Stanford University, Stanford, CA, 94305, USA
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Lawrence, Brian. A density result for real hyperelliptic curves. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1219-1224. doi : 10.1016/j.crma.2016.10.014. http://www.numdam.org/articles/10.1016/j.crma.2016.10.014/

[1] Bogatyrev, A. Extremal Polynomials and Riemann Surfaces, Springer, Berlin, 2012

[2] Robinson, R.M. Conjugate algebraic integers in real point sets, Math. Z., Volume 84 (1964), pp. 415-427

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