Singularities of the union of two smooth curves I am looking for a reference that describes what are the possible singularities of a curve $C=C_1 \cup C_2$ which is the union of two smooth irreducible curves $C_1,\,C_2$ on a smooth surface.
Doing some computations, my guess is that these singularities are always of type $A_{2m-1}$ ($m\in \mathbb{N}^*$), but I would like to see a clear proof.
 A: I figured out what @Mohan told me and wrote a proof
Let $C_{1},\,C_{2}$ be two smooth curves on a smooth surface and
let $p$ be an intersection point of $C_{1}$ and $C_{2}$. One can
suppose that there are local coordinates $x,y$ around $p$ such that
$C_{1}$ is given by $y=0$. 
If the intersection is transverse, up to a change of variables, one
can suppose that $C_{2}$ has equation $x=0$. Then $xy=0$ is the
equation of a $\frak{a}_{1}$ singularity. 
Otherwise the equation of $C_{2}$ is given by $y+g(x,y)=0$ in the
completion of the local ring at $p$, where 
$$
g=\sum_{r\geq2}P_{r}(x,y)
$$
 is a formal series in $x,y$ such that $P_{r}(x,y)$ is homogeneous
of degree $r$ (or $0$).  By the implicit function theorem, there
exists an holomorphic function $h$ such that $h(x)+g(x,h(x))=0$
for all $x$ in a neighborhood of $0$. In other words, locally around
$p$, the curve $C_{2}$ is the graph of $x\to(x,h(x))$. 
There exists an integer $m>0$ and an holomorphic function $q$ such
that $h(x)=x^{m}q(x)$ with $q(0)\neq0$. We choose a determination
of $\sqrt[m]{q(x)}$ and define the map (locally around $(0,0)$)
$$
\phi:(x,y)\to(x\sqrt[m]{q(x)},y).
$$
Then $\phi(C_{2})=C_{2}'$ where $C_{2}'=\{(x,y)\,|\,y=x^{m}\}$ and
$\phi(C_{1})=C_{1}$ (near $(0,0)$). The map $\phi$ is (locally near $(0,0)$) a
biholomorphism. Therefore the singularity at the origin of $C_{1}+C_{2}$
is the same as the one on $C_{1}+C_{2}'$ at the origin. One thus
can suppose $C_{2}=\{y=x^{m}\}$. 
Since $C_{1}=\{(x,y)\,|\,y=0\}$, the singularity at $p$ of $C_{1}+C_{2}$
has equation $y(y-x^{m})=0$. By the change of variable $z=y-\frac{1}{2}x^{m}$,
the equation $y(y-x^{m})=0$ becomes 
$$
(z+\frac{1}{2}x^{m})(z-\frac{1}{2}x^{m}),
$$
and performing another change of variables, one gets the equation
$$
y^{2}-x^{2m}=0,
$$
which is the equation of a $\frak{a}_{2m-1}$ singularity. 
It would be nice to do understand what is valid for any fields...
