On the 1-pointed curves arising as etale covers of the affine line in positive characteristic

Details On the 1-pointed curves arising as etale covers of the affine line in positive characteristic On the 1-pointed curves arising as etale covers of the affine line in positive characteristic

2003-09-23

arxiv.org/abs/math/0309386v1

Mathematics

Algebraic Geometry

19 pages

Scientific paper

Let k be an algebraically closed field of positive characteristic. The goal of this paper is to characterize the proper smooth curves X/k of positive genus g equipped with a k-rational point P such that X\P can be realized as an etale cover of the affine line. A first elementary analysis shows that such a cover exists if and only if there exists an exact regular differential form on X having a unique zero at P (of order 2g-2). The main inconvenient of this approach is that it doesn't give any control on the degree of the cover. A finer investigation, essentially based on the duality between the Cartier operator and the Frobenius map allows to overcome this problem.

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