The genus of a curve of Fermat type

2007/10/22 by Jeremiah M. Kermes, Kermes, Jeremiah M.
Mathematics · #14H45 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H45

paper · pdf · doi:10.48550/arxiv.0710.3890

12 pages

arxiv created 2007/10/22 · arxiv updated 2009/12/01

Abstract

In this paper we begin to study curves on a weighted projective plane with one trivial weight, \mathbb P(1,m,n), by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to the one from Fermat's last theorem. We begin by finding local coordinates for the standard affine cover of the plane, and then prove that the curve is smooth. This is done by pulling the curve up to the surface's desingularization. Then a map from the curve to \mathbb P¹ is constructed, and it's ramification divisor is determined. We conclude by applying Hurwitz's theorem to this map to obtain C's genus.

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