# Is there an invariant similar to the delta invariant that distinguishes an $A_2$ node form an $A_1$ node?

Consider the following question: If two nodes collide what do you get? First of all it can not be a strict $A_2$ node, because the delta invariant of that is $1$. So it has to be more singular than an $A_2$ node. It can be an $A_3$ node because the delta invariant of that is $2$.

Is there any simple argument to show that if an $A_2$ node and an $A_1$ node collide, then we can not get a strict $A_3$ node? The delta invariant doesn't help. Is there some other invariant that can answer this question? Note that I am NOT asking what do we actually get when an $A_2$ node and an $A_1$ node collide. I merely want to show that we can not get a strict $A_3$ node.

-

A miniversal deformation of the $A_3$ singularity is given by the family $y^2 = a + bx + cx^2 + x^4$. There is no member in this family with nodes of type $A_1$ and $A_2$ so it follows that we cannot get an $A_3$ node from collisions of two nodes of type $A_1$ and $A_2$.
The singularities in the family correspond to multiple roots of the polynomial $a + bx + cx^2 + x^4$. Since this is of degree $4$, it can never have both a double root and a triple root. – ulrich Nov 17 '11 at 15:16