Let $X\subset \mathbb P^n_{\mathbb C}$ be a closed algebraic subset and $x\in X$. Suppose tha tangent cone of $X$ at $x$ is the union of (say) a plane and a line (meeting at the origin). Can we conclude that $X$ is around (in the euclidian/étale topology) $x$ a union of a smooth surface and a smooth curve meeting transversally?

$\begingroup$ If you give the tangent cone its natural scheme structure, there is a flat specialization of $X$ to the tangent cone. Thus, if the tangent cone is reduced, then also $X$ is (locally) reduced, cf. Theoreme 12.1.1.(viii) of EGA IV_3. Similarly, the union of all topdimensional components of $X$ have HilbertSamuel multiplicity 1 at $x$, so that union is smooth at $x$. I am not sure about the rest; maybe check PieneSchlessinger. $\endgroup$– Jason StarrJan 30, 2019 at 14:36

$\begingroup$ @Jason Starr: what is it meant by a flat specialization? Is it a flat map $\mathcal{X}\to \mathrm{Spec}(\textrm{dvr})$ such that the general fiber is $X\times_{\mathbb{C}} K$ (where $K=\mathrm{frac}(\textrm{dvr}$) ) and the special fiber is the tangent cone? $\endgroup$– QfwfqJan 30, 2019 at 15:28

$\begingroup$ @Qfwfq Yes, that is exactly what I meant. $\endgroup$– Jason StarrJan 30, 2019 at 15:46
1 Answer
Because of the flat specialization Jason mentioned, the dimension of the topdimensional component of $X$ must equal the dimension of the topdimensional component of the tangent cone. So the topdimensional component of $X$ (near $x$) is a smooth surface. Let $S$ be this surface. We have an exact sequence of modules $0 \to K \to \mathcal O_X \to \mathcal O_S \to 0$ where $K$, the kernel, is supported in dimension $\leq 1$. We can take the associated graded to the degree filtrations on each of these (an equivalent construction of the tangent cone) to obtain an exact sequence, where the middle term is the ring of functions on the tangent cone of $X$ and the right term is the ring of functions on the tangent cone of $S$, which is a plane. So the associated graded of $K$ is the module of functions on the line which vanish at the origin. So $K$ has HilbertSamuel multiplicity one and is the sheaf of functions of a smooth curve, at least away from finitely many points. The tangent cone of this curve is transverse to the tangent cone of $S$. So $X$ is the union of a smooth surface and a smooth curve, intersecting transversely at $x$, away from something zerodimensional. But the tangent cone of this union of the surface and the curve is the union of a plane and a line, so there is no room to hide any additional points.
Then because $X$ is Zariskilocally the union of a smooth surface and a smooth curve intersecting transversely, it is etalelocally the union of a plane and a line.