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Automorphism groups of smooth plane curves. (English) Zbl 1433.14024
Let $$C$$ be a smooth plane algebraic curve defined over the complex number field. The automorphism group of $$C$$ has been the subject of several studies in algebraic geometry (see [K. Oikawa, Kōdai Math. Semin. Rep. 8, 23–30 (1956; Zbl 0072.07702)] and [T. Arakawa, Osaka J. Math. 37, No. 4, 823–846 (2000; Zbl 0981.30029)] for instance) and, although there are examples for which this group is completely known (see [P. Tzermias, J. Number Theory 53, No. 1, 173–178 (1995; Zbl 0853.14015)], [T. Harui et al., Kodai Math. J. 31, No. 2, 257–262 (2008; Zbl 1145.14027)] and [T. Harui et al., Kodai Math. J. 33, No. 1, 164–172 (2010; Zbl 1192.14028)] for instance), there has not been a general result on its structure.
In the present work, the author addresses the following three problems:
(1)
Classify the group of automorphisms of smooth plane algebriac curves;
(2)
Give a sharp upper bound for the order of the group of automorphisms of such curves;
(3)
Determine smooth plane algebraic curves with automorphism group of large order
and an answer to each of them is provided. More specifically, it is shown that smooth plane algebraic curves defined over the complex number field can be divided into the following five classes according to their full group of automorphisms:
(a-i)
Curves whose full automorphism group is cyclic.
(a-ii)
Curves whose full automorphism group is the central extension of a finite subgroup of Möbius group $$\text{PGL}(2,\mathbb{C})=\text{Aut}(\mathbb{P}^{1})$$ by a cyclic group.
(b-i)
Curves descendants of Fermat curves.
(b-ii)
Curves descendants of Klein curves.
(c)
Curves for which their full automorphism group is isomorphic to a primitive subgroup of $$\text{PGL}(3,\mathbb{C})$$.
As a consequence, stronger upper bounds for the order of such groups are obtained, and a classification of the exceptional cases is provided.

##### MSC:
 14H37 Automorphisms of curves 14H50 Plane and space curves 14H45 Special algebraic curves and curves of low genus
##### Keywords:
automorphism groups; plane curves
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