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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.

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  • $\begingroup$ What is $A_n$ singularity for you? $\endgroup$
    – Mohan
    Commented Apr 1, 2018 at 15:43
  • $\begingroup$ @Mohan : $A_n$ singularity has equation $y^2-x^{n+1}=0$ $\endgroup$
    – Xavier49
    Commented Apr 1, 2018 at 17:04
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    $\begingroup$ At any point $P\in C_1\cup C_2$, in the completion of the local ring of the surface, you can easily see that $C_1\cup C_2$ is given by $x(x-y^n)=0$ for suitable choices of parameters $x,y$. So, if characteristic is not 2, this can be changed to $x^2-y^{2n}=0$ which is what you are claiming. $\endgroup$
    – Mohan
    Commented Apr 1, 2018 at 18:35
  • $\begingroup$ @Mohan Thanks! Yes, but the "easily" part is not so easy for me, because I do not know how to tell when too singularities are 'the same', I mean I suppose there are some equivalence relations but I do not know which they are. By example one has the singularity $(x-y^a)(x+y^b)=0$, with say $a \leq b$. By expanding that polynomial, my intuition tells me that this is a $A_{2a}$ singularity, but I would like to understand rigorously why. More generally with e.g. $(x+vy^a)(x+uy^b)=0$, where $u,v$ are non equal constants. $\endgroup$
    – Xavier49
    Commented Apr 2, 2018 at 7:00

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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...

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