Does anyone have a reference for the universal deformation space of a cuspidal plane cubic curve? Specifically, a reference that discusses its discriminant locus -- Apparently it has a cuspidal discriminant.
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In Moduli of Curves by Harris-Morrison, page 97, it says
The space of first-order deformation of a singular point $p$ of a plane curve $C\subseteq \mathbb A^2$ given by $f(x,y)=0$ is the local ring of $C$ at $p$ modulo the Jacobian ideal $\mathcal{J}$ generated by the partial derivatives $\partial f/\partial x$ and $\partial f/\partial y$.
It turns out that the universal deformation space of the cuspidal plane curve $y^2=x^3$ is $\mathbb A^2$ with coordinate $(a,b)$ and the universal family is $$y^2=x^3+ax+b.$$
A straightforward computation shows that the discriminant locus is $\Delta=\{27b^2+8a^3=0\},$ (which is again a plane cuspidal curve), i.e., if $(a,b)\in \Delta\setminus\{0\}$, then the curve is nodal, and the curve will be smooth if $(a,b)\notin \Delta$.
answered Aug 21, 2020 at 23:56