Rational curves on Fermat hypersurfaces

2011/11/24 by Mingmin Shen, Shen, Mingmin · 1 citation
Arts and Humanities · Mathematics · Social Sciences · #14C05 #14D10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1111.5657

openalex publication_date 2011/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we study rational curves on degree p^r+1 Fermat hypersurface in \PP^pr+1_k, 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 charateristic 0. We show that if there exists N₀ such that for all e≥ N₀ there is a degree e very free rational curve on X, then N₀> p^r(p^r-1).

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