Simply branched covers of curves and wild conductor exponents

We use deformation theory to show that a cover of smooth, projective, geometrically connected curves $\pi \text{\hspace{0.17em}⁣}:C\to D$ over a $p$-adic field can be $p$-adically perturbed to obtain a nearby simply branched cover ${\pi }^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:C^{\mathrm{\prime }}\to D$ and that, if $D={\mathbb{P}}^1$ and ${\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ is very ample, then we may take $C^{\mathrm{\prime }}=C$. As an application, we give a formula for the wild conductor exponent of a curve at a prime $p>d$ in terms of the ramification data of any degree $d$ cover $C\to {\mathbb{P}}^1$.