Suppose that $C$ and $D$ are curves of finite type over an algebraically closed field $k$ (we make some of these hypotheses for simplicity). We view these as pointed curves with singularities $c \in C$ and $d \in D$. I'm happy to assume that these singularities look analytically locally like $n$-coordinate-lines through the origin in $\mathbb{A}^n$; in other words $k[x_1, \dots, x_n]/(\dots, x_i x_j, \dots)$.

If you'd like to start with simple (planar) nodes (ie, $k[x,y]/(xy)$), that's ok too.

Further suppose that there is a finite map $C \to D$ taking $c$ to $d$.

**Question**: Is there an accepted notion of what it means for such a map to have tame ramification at $d$?

Particularly, has it been studied before, and if so, where can I read about it?