Quotient curves of smooth plane curves with automorphisms.

*(English)*Zbl 1192.14028Let \(C\) 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.

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.

Reviewer: Aaron Wootton (Portland)

##### MSC:

14H51 | Special divisors on curves (gonality, Brill-Noether theory) |