Hypersurface of singular plane cubics In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The degree of $X$ is $12$.
Let $SX\subset X$ be the singular locus of $X$. Are the dimension and the degree of $SX$ and the type of singularity of $X$ at a general point of $SX$ known?
Thanks.
 A: Let $X = v_3(\mathbb{P}^2) \subset \mathbb{P}^9$ be the third Veronese embedding of $\mathbb{P}^2$ (given by the global sections of $\mathcal{O}_{\mathbb{P}^2}(3)$). A hyperplane section of $X$ corresponds to the vanishing locus of section of $\mathcal{O}_{\mathbb{P}^2}(3)$, that is a plane cubic in $\mathbb{P}^2$. The variety of singular hyperplane sections of $X$ is called the projective dual of $X$ and denoted by $X^* \subset \left(\mathbb{P}^9\right)^*$. It parametrizes the singular plane cubics. Its singular locus is not irreducible. A union of some of its components parametrizes plane cubics with (at least) a cusp and the union of the other parametrizes plane cubics with (at least) two singular points.
An easy count of dimension shows that both these schemes are hypersurfaces in $X^*$ (and in particular $X^*$ is not a normal hypersurface). On the other hand, given any smooth $X \subset \mathbb{P}^n$, there is a general theory studying the subschemes of $X^*$ which parametrize cuspidal and multinodal hyperplane sections. Using the Reflexivity Theorem in projective duality, you can show that, under favorable hypotheses, they are always hypersurfaces in $X^*$. Holweck's Phd thesis, and in particular chapter 2, is a good reference for this theory.
Edit : As Sasha suggests in his comment, the conormal space $\mathbb{P}(N_{X/\mathbb{P}^{3}}(-1))$ is a resolution of singularities. Furthermore, since the fiber of the map $ \mathbb{P}(N_{X/\mathbb{P}^{3}}(-1)) \longrightarrow X^*$ over $H$ is the scheme-theoretic singular locus of the curve corresponding to $H$, on can deduce that the fibers over generic points of $X^*_{cusp}$ and $X^{*}_{multinode}$ are respectively:
one fat point ($A_2$) and two simple points ($2A_1$). This suggests that all generic points in $X^*_{cusp}$ and $X^*_{multinode}$ have multiplicity $2$ in $X$. In particular, in order to compute the genus of a generic plane section of $X$, one just needs to compute the degree the reduced hypersurface components of $X^*$.
