Let $k$ be an algebraically closed field. I would like to prove that there are only two possible $k$-scheme structures on a triple point, namely that of a $2$-jet $\textrm{Spec }k[x]/(x^3)$, and that of $\textrm{Spec }k[x,y]/(x^2,xy,y^2)$.
A triple point "is" a local Artin $k$-algebra $(A;\mathfrak m)$ of dimension $3$. So I have to show that $A$ is either isomorphic to a $2$-jet ring, or to $k[x,y]/(x^2,xy,y^2)$. I do not know how to do it. The only thing I know is that $A/\mathfrak m=k$, so by $\dim_k A=3$ we see that as a $k$-vector space, $\mathfrak m$ has dimension $2$. If I assume $\mathfrak m$ is generated by "vectors" $x,y$, then (since $\mathfrak m^2=0$) we have relations $x^2=0,xy=0,y^2=0$. This is not precise at all, though.
There is a sort of intuition that I cannot make precise. It is as follows: I have a supporting point $[\mathfrak m]\in\textrm{Spec }A$ and I have to give two tangent vectors; I have essentially two choices: either I choose them to be linearly independent, or linearly dependent. Can I somehow make this precise and get the result?
Thanks in advance.
