First steps towards total reality of meromorphic functions

2006/05/02 by T. Ekedahl, B. Shapiro, Ekedahl, T. +3
Mathematics · #14P05 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14P05

paper · pdf · doi:10.48550/arxiv.math/0605075

10 pages, 1 figure

arxiv created 2006/05/02 · arxiv updated 2009/12/01

Abstract

It was earlier conjectured by the second and the third authors that any rational curve g:\mathbb CP¹→ \mathbb CPⁿ such that the inverse images of all its flattening points lie on the real line \mathbb RP¹⊂ \mathbb CP¹ is real algebraic up to a linear fractional transformation of the image \mathbb CPⁿ. (By a flattening point p on g we mean a point at which the Frenet n-frame (g',g'',...,g⁽ⁿ⁾) is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees 2,3 and several other cases.

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