Algebraic curves with a large non-tame automorphism group fixing no point

Giulietti, M.; Korchmáros, G.

doi:10.1090/S0002-9947-2010-05025-1

Algebraic curves with a large non-tame automorphism group fixing no point
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by M. Giulietti and G. Korchmáros

Trans. Amer. Math. Soc. 362 (2010), 5983-6001

DOI: https://doi.org/10.1090/S0002-9947-2010-05025-1

Published electronically: June 10, 2010

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Abstract:

Let $\mathbb {K}$ be an algebraically closed field of characteristic $p>0$, and let $\mathcal {X}$ be a curve over $\mathbb {K}$ of genus $g\ge 2$. Assume that the automorphism group $\mathrm {Aut}(\mathcal {X})$ of $\mathcal {X}$ over $\mathbb {K}$ fixes no point of $\mathcal {X}$. The following result is proven. If there is a point $P$ on $\mathcal {X}$ whose stabilizer in $\mathrm {Aut}(\mathcal {X})$ contains a $p$-subgroup of order greater than $gp/(p-1)$, then $\mathcal {X}$ is birationally equivalent over $\mathbb {K}$ to one of the irreducible plane curves (II), (III), (IV), (V) listed in the Introduction.

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