are pure.

A point $p \in C$ is called a [break for the morphism $f$]{} (or simply a [break]{}), if the sheaf ${\mathcal C}_f$ is not locally free at $p$. We say that the morphism $f$ has no breaks if the sheaf ${\mathcal C}_f$ is locally free.

It is clear from the definition that a smooth point of $C$ is never a break.

Suppose the morphism $f$ is finite. A point $p \in C$ is called a [ramification point]{}, if it belongs to the support of the sheaf ${\mathcal Q}_f$. We call [ ramification divisor of $f$]{} the (Weil) divisor whose multiplicity at $p\in C$ is the length of ${\mathcal Q}$ at $p$.

Let $f_1 : C_1 \rightarrow X$ and $f_2 : C_2 \rightarrow X$ be non-constant morphisms from two smooth curves to a smooth surface. Suppose $p_1 \in C_1$ and $p_2 \in C_2$ are points such that $f_1 (p_1) = f_2 (p_2) = q$, let $u$ and $v$ be local coordinates on $X$ near $q$ and let $x_1$ and $x_2$ be local parameters for $C_1$ and $C_2$ near $p_1$ and $p_2$ respectively. Since $f_1$ and $f_2$ are not constant, there exist integers $k_1$ and $k_2$ such that $$ f\_1 