# Smooth complete intersections

Let $$X_{2,3}\subset\mathbb{P}^n$$, with $$n\geq 5$$, be a complete intersection of a quadric $$X_2$$ and a cubic $$X_3$$ containing a $$2$$-plane $$H$$. Assume $$X_2$$ and $$X_3$$ to be general among the hypersurface of the same degree containing $$H$$. In particular $$X_2$$ and $$X_3$$ are smooth.

Question: If $$n = 5$$ is $$X_{2,3}$$ necessarily singular? Is $$X_{2,3}$$ smooth for $$n\geq 6$$?

If $$X \subset \mathbb{P}^n$$ is a non-degenerate, smooth complete intersection variety of dimension at least $$3$$, then the restriction map $$\operatorname{Pic}(\mathbb{P}^n) \to \operatorname{Pic}(X)$$ is an isomorphism by Grothendieck-Lefschetz theorem. So, we get $$\operatorname{Pic}(X) = \mathbb{Z}$$, generated by the hyperplane section.
This implies that $$X$$ contains no linear spaces of codimension $$1$$; in particular, the threefold $$X_{2,3} \subset \mathbb{P}^5$$ is necessarily singular.
If $$n=5$$ then let $$\mathbb P^5$$ have coordinates $$x_0,\ldots,x_5$$ and suppose the plane is $$H=\mathbb P^2_{(x_0:x_1:x_2)}$$. The two equations of $$X$$ are necessarily of the form $$\begin{pmatrix} A_1 & B_1 & C_1 \\ D_2 & E_2 & F_2 \end{pmatrix}\begin{pmatrix} x_3 \\ x_4 \\ x_5 \end{pmatrix} = 0$$ for some polynomials $$A_1,\ldots,F_2\in\mathbb C[x_i]$$ of the indicated degree. The scheme $$Z\subset \mathbb P^5$$ defined by the $$2\times 2$$ minors of the $$2\times 3$$ matrix has codimension 2, and thus $$Z\cap H$$ is nonempty (generically given by some points). Since both equations have order of vanishing $$\geq2$$ along $$(Z\cap H)\subset X$$, the 3-fold $$X$$ must be singular there.
In dimension $$n\geq6$$ the scheme $$Z$$ (obtained by the analogous argument) has codimension $$n-3\geq3$$ so in the general case $$Z\cap H=\emptyset$$ and the $$n$$-fold $$X$$ is smooth (along $$H$$ at least). Essentially, since the matrix never drops rank we can always use the equations to eliminate two of the variables $$x_3,\ldots,x_n$$, showing that $$X$$ is smooth of dimension $$n$$ at every point along $$H$$.
• It's not true that both equations have order of vanishing $\geq 2$ at these points but rather that some linear combination does (but this still implies a singularity). Jan 18 at 12:58