Deformations of a cuspidal plane cubic

To practice with deformations, I am trying to compute the space of first order deformations of the cuspidal curve $X=\textrm{Spec }B$, where $B=P/I$, $P=k[x,y]$ and $I=(f)=(y^2-x^3)$.

The conormal sequence $$I/I^2\overset{d}{\longrightarrow}\Omega_{P/k}\otimes_PB\cong B\,dx\oplus B\,dy\longrightarrow \Omega_{B/k}\longrightarrow 0$$ induces a $B$-linear map (the "dual" of $d$) $$\alpha:\hom_B(\Omega_{P/k}\otimes_PB,B)\to\hom_B(I/I^2,B).$$ Just to be as clear as possible with those who read this question, I give a list of "synonyms" of the space of first order deformations of $X$ (hoping not to confuse anyone, nor to state something wrong!):

• $\textrm{coker } \alpha$,
• $T^1(B/k,B)$,
• $\textrm{Ex}_k(B,B)$,
• $\textrm{Ext}^1_B(\Omega_{B/k},B)$.

I attempted to determine $\textrm{coker }\alpha$, but now I'm stuck.

What I did was just to translate the maps explicitly. So for instance $I/I^2$ is a principal module generated by $\overline f=f+I^2$, so $$d:\overline f\mapsto 2y\,dy-3x^2\,dx.$$ This is useful to determine $\alpha$. The source is generated by two vector fields $\partial/\partial x$ and $\partial/\partial y$, so: \begin{align} \alpha: &\partial/\partial x\mapsto (\overline f\mapsto \partial f/\partial x=-3x^2),\\ \alpha: &\partial/\partial y\mapsto (\overline f\mapsto \partial f/\partial y=2y). \end{align}

Now, as $I/I^2$ is a principal module, we get an identification $(\star)$ $$\textrm{coker }\alpha=\hom_B(I/I^2,B)/\textrm{Im }\alpha\overset{(\star)}{\cong} B/(-3x^2,2y)\cong k[t^2,t^3]/(t^4,t^6).$$

I cannot go further, and I remember having read on Moduli of Curves that this space should be $2$-dimensional. Can anyone help me to conclude?

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1 Answer

The quotient $B/(-3x^2,2y) \cong B/(x^2,y) = k\langle 1, x \rangle$.

A mini-versal deformation is given by $x^3 + y^2 + ax + b$.

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