Automorphism groups of smooth plane curves.

*(English)*Zbl 1433.14024Let \(C\) 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:

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

- (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 \(\text{PGL}(3,\mathbb{C})\).

Reviewer: Mariana Coutinho (São Carlos)