Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $X$ be a smooth projective connected variety over $\mathbf C$ with infinitely many etale covers.

If $\dim X =1$, this holds if and only if the genus of $X$ is positive.

Do we have a similar criterion in dimension two?

Clearly, if $\dim X=2$ this holds if $X$ is an abelian variety,but not if it is a K3 surface. Also, if $X$ is a Jacobian elliptic fibration, we have infinitely many finite degree topological covers.

What if $X$ is of general type? That is, does a surface of general type have infinitely many etale covers?

Note that the topological fundamental group of $X$ is finitely generated, thus the existence of infinitely many etale covers exists that the degree of such an etale cover is not bounded.

share|cite|improve this question
up vote 1 down vote accepted

A surface of general type could be simply connected, contrary to the situation for ''curves of general type''. For instance, consider the Fermat surface $x_0^5+x_1^5+x_2^5+x_3^5 =0$ in $\mathbf P^3$. It is a smooth complete intersection of degree $5$, thus it is simply connected. It is of general type because its Kodaira dimension is easily seen to equal two (once you write down the canonical divisor).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.