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Quotient curves of smooth plane curves with automorphisms. (English) Zbl 1192.14028
Let $$C$$ be a smooth plane curve over $${\mathbb C}$$ of degree $$d\geq 4$$ and suppose that $$\sigma$$ is an automorphism of $$C$$ of order $$n\geq 2$$ with number of fixed points equal to $$f$$. Then $$\sigma$$ extends to an automorphism $$\tilde{\sigma}$$ of $${\mathbb P}^2$$ which can be represented by a $$3\times 3$$ diagonal matrix with either two $$1$$’s on the diagonal (called a type 1 automorphism) or a single $$1$$ on the diagonal (called a type 2 automorphism). In the paper under review, the authors determine some properties of the quotient curve $$B=C/\langle \sigma \rangle$$, the automorphism $$\sigma$$ and the degree $$d$$ for type 1 and 2 automorphisms.
For type $$1$$ automorphisms, the authors prove a number of extensive results which hold for any $$n$$. For example, the authors show that $$d\equiv 0$$ or $$1\mod{(n)}$$ and moreover when $$d\equiv 0\mod{(n)}$$ we have $$f=d$$, and when $$d\equiv 1\mod{(n)}$$, $$f=d+1$$. Additionally they show that the quotient curve $$B$$ is isomorphic to a curve of degree $$d$$ in $${\mathbb P}^{n+1}$$ attaining the largest geometric genus given by the Castelnuovo bound. They also prove a converse statement. Specifically, given a curve $$B$$ satisfying the properties derived, and integers $$n\geq 2$$ and $$d\equiv 0$$ or $$1\mod{(n)}$$, there always exists a smooth plane curve $$C$$ and an automorphism $$\sigma$$ of order $$n$$ of type 1 and a cyclic covering $$\pi : C\rightarrow B=C/\langle \sigma \rangle$$.
For type 2 automorphisms, the authors are able to provide similar partial results in the special case that $$n=p$$ a prime. Specifically, they show that either $$f=0$$ and $$d\equiv 0\mod{(p})$$, $$f=2$$ and $$d\equiv 1$$ or $$2\mod{(p})$$, or $$f=3$$ and $$d^2-3d+3\equiv 0\mod{(p})$$ and $$p\equiv 1\mod{(6)}$$ or $$p=3$$. They finish by giving specific examples to show that each of these three cases occur.

##### MSC:
 14H51 Special divisors on curves (gonality, Brill-Noether theory)
##### Keywords:
plane curves; automorphisms; covering; quotient curves; linear system
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