Rational pullbacks of Galois covers

Abstract

The finite subgroups of ${\mathrm{PGL}}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\to {\mathbb{P}}_{\mathbb{C}}^1$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps ${\mathbb{P}}_{\mathbb{C}}^1\to {\mathbb{P}}_{\mathbb{C}}^1$. For $G\subset {\mathrm{PGL}}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\not\subset {\mathrm{PGL}}_2(\mathbb{C})$, letting the branch point number grow provides truly new Galois realizations $F/\mathbb{C}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of ${\mathbb{P}}_{\mathbb{C}}^1$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset {\mathrm{PGL}}_2(\mathbb{C})$.