Define the surface $X$ to be the total space of $\mathcal{O}\_{\mathbb{P}^1}(-5)$.
By contracting the exceptional curve in $X$, we get a surface with an isolated singularity. I am looking for the equation (or the set of equations) that describes this singularity (as a surface in some $\mathbb{C}^n$, possibly just $\mathbb{C}^3$).


For example in the case of $X$ being the total space of $\mathcal{O}\_{\mathbb{P}^1}(-2)$,
the resulting singularity is the $A\_1$ singularity given by
$$x^2+yz=0$$.



Which isolated surface singularity comes from a -5 curve?
The weighted projective plane $\mathbb{P}(1,1,n)$ can be viewed as $\mathbb{P}(1,1,n)=\mathbb{C}^3\setminus \{0\}/(x,y,z)\sim (\lambda x,\lambda y,\lambda^n z)$. For $n=1$ we obtain the standard projective plane. For $n>1$, the point $(0,0,1)$ is the unique singular point, and the blow-up of this point is a Hirzebruch surface $\mathbb{F}\_n$, with exceptional divisor $E\simeq \mathbb{P}^1$ of self-intersection $-n$. Hence, what you are looking for is just the quotient singularity of $\mathbb{P}(1,1,n)$.


If you want a local embedding into some affine space, take the local embedding $(x,y,z)\mapsto (x^n/z,x^{n-1}y/z,\dots,xy^{n-1}/z,y^n/z)$. The equations are probably easy to obtain from this explicit description. For $n=2$ you obtain what you already described.


