3
$\begingroup$

Let $X\to \mathbf{P}^1$ be a branched cover of the complex projective line, where $X$ is a compact connected Riemann surface. Let $G=\mathrm{Aut}(Y/\mathbf{P}^1)$.

Question 1. Could somebody provide some examples of branched covers $X\to \mathbf{P}^1$ for which $G= (0)$?

One could construct a compact connected Riemann surface without automorphisms to this end.

1 Answers 1

2

As you suggest, it is enough to prove the existence of a compact Riemann surface $X$ with only the identity as automorphism .
Indeed, $X$ has a non constant meromorphic function $f$ and by considering it as a ramified covering $f:X\to \mathbb P^1$ we get our example .

Actually examples are plenty: a generic compact Riemann surface of genus $g\geq 3$ has no automorphism besides the identity . This is a theorem proved by by Baily in 1961.
You can get a feel for that theorem by going to this MathOverflow question, to which Torsten Ekedahl (whom we sadly lost little ago) gave a great answer.