Algebraic Geometry
Rational curves on Fermat hypersurfaces
Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 781-784.

In this note we study rational curves on degree pr+1 Fermat hypersurface in Pkpr+1, where k is an algebraically closed field of characteristic p. The key point is that the presence of Frobenius morphism makes the behavior of rational curves to be very different from that of characteristic 0. We show that if there exists N0 such that for all eN0 there is a degree e very free rational curve on X, then N0>pr(pr1).

Note nous étudions les courbes rationnelles sur les hypersurfaces de Fermat de degré pr+1 dans Pkpr+1, où k est un corps algébriquement clos de caractéristique p. Le point essentiel est la présence du morphisme de Frobenius qui rend le comportement des courbes rationnelles très différent du cas de caractéristique 0. Nous montrons que si N0 est un entier tel que pour tout eN0 il y ait une courbe rationnelle très libre de degré e sur lʼhypersurface de Fermat, alors N0>pr(pr1).

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Published online:
DOI: 10.1016/j.crma.2012.09.015
Shen, Mingmin 1

1 DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK
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Shen, Mingmin. Rational curves on Fermat hypersurfaces. Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 781-784. doi : 10.1016/j.crma.2012.09.015. http://www.numdam.org/articles/10.1016/j.crma.2012.09.015/

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