tangent cone and local picture

Question

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?

Answer 1

Because of the flat specialization Jason mentioned, the dimension of the top-dimensional component of $X$ must equal the dimension of the top-dimensional component of the tangent cone. So the top-dimensional 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 Hilbert-Samuel 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 zero-dimensional. 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 Zariski-locally the union of a smooth surface and a smooth curve intersecting transversely, it is etale-locally the union of a plane and a line.