Abstract:
Given a branched cover $p:\widetilde{\Sigma}\to\Sigma$ beteween closed orientable surfaces, the famous Riemann-Hurwitz formula
relates the Euler characteristics of $\widetilde{\Sigma}$ and $\Sigma$, the total degree $d$ of $p$,
the number $n$ of branch points in $\Sigma$ and the sum of the lengths of
the partitions $\left(\left(d_{i,j}\right)_{j=1}^{m_i}\right)_{i=1}^n$
of $d$ given by the local degrees of $p$ at the preimages of the branch points.
The Hurwitz existence problem asks whether a given combinatorial datum
$$\left(\widetilde{\Sigma},\Sigma,d,n,\left(\left(d_{i,j}\right)_{j=1}^{m_i}\right)_{i=1}^n\right)$$
satisfying the Riemann-Hurwitz formula is actually realized by a branched cover $p:\widetilde{\Sigma}\to\Sigma$.
The answer is now known to be always in the affirmative when $\Sigma$ has positive genus, but not
when $\Sigma$ is the Riemann sphere. I will report on recent progress on the problem based on a connection
with the geometry of 2-orbifolds.
The talk is based on the joint papers with with M. A. Pascali [1] and [2].
References:
M. A. Pascali, C. Petronio, Surface branched covers and geometric $2$-orbifolds. Trans. Amer. Math. Soc. 361 (2009), 5885–5920.
M. A. Pascali, C. Petronio, Branched covers of the sphere and the prime-degree conjecture. Ann. Mat. Pura Appl. 191 (2012), 563–594.
