  We study group actions on regular models of curves. If $X$ is a smooth curve
defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely
ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on
$X_{K'}$. In this paper we study the extension of this $G$-action to certain
regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer
trace of the endomorphism induced by a group element on the alternating sum of
the cohomology groups of the structure sheaf of the special fiber of such a
regular model. Inspired by this global study, we also consider similar
questions for Galois actions on the exceptional locus of a tame cyclic quotient
singularity.
  We apply these results to study a natural filtration of the special fiber of
the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes.
We show that the jumps in this filtration only depend on the fiber type of the
special fiber of the minimal regular model with strict normal crossings for $X$
over $\Spec(R)$, and in particular are independent of the residue
characteristic. Furthermore, we obtain information about where these jumps
occur. We also compute the jumps for each of the finitely many possible fiber
type for curves of genus 1 and 2.
