Is the sum of a radical ideal and the ideal of a generic linear space intersecting that ideal radical? Let $X \subseteq \mathbb{C}^n$ be an irreducible algebraic set that forms a cone, and let $I=I(X) \subseteq \mathbb{C}[x_1,...,x_n]$. Let $m < n$ and $k\leq m$ be positive integers. Is it true that for a generic collection of elements $v_1,..., v_k \in X$ and $v_{k+1},..., v_m \in \mathbb{C}^n$, it holds that $I + I(\text{span}\{v_1,...,v_m\})$ is radical?
 A: In general no. I'm only familiar with this question with $k=0$. Here are two basic counter-examples (both with $k=0$)
(1) Take $X$ singular and dimension $n-m$, so we are intersecting $X$ with a generic codimension $n-m$-plane. Then the intersection is a point of multiplicity equal to the multiplicity of $0$ as a singular point of $X$.
(2) Let $X$ be non-Cohen Macaulay with depth $\delta < \dim X$. Then intersecting $X$ with a codimension $\delta$ plane gives a variety with an embedded point.
If we impose that $X$ is Cohen-Macaulay, that $k=0$ and that $n-m < \dim X$, then the answer is "yes". Recall Serre's Theorem that "reduced" is equivalent to R0 and S1. $X$ Cohen-Macaulay means that $X$ is $S_{\dim X}$, and $X$ reduced means that $X$ is $R_0$. Taking $q$ generic hyperplane sections (through the singularity of $X$) turns this into $S_{\dim X - q}$ and $R_0$, so still reduced. I can't quickly find good references for this; I'm willing to try harder if someone else doesn't beat me too it.
The problem with $k>0$ seems more interesting to me, and I haven't seen it before.
A: I think another source of counterexamples might be the Harshorne-Hirschowitz theorem (lifted from this paper):
Theorem (Hartshorne-Hirschowitz 1981):  Let $X\subset\mathbb{P}^n, \ (n\geq 3)$ be a generic union of $e$ lines.  Then $X$ has good postulation, i.e.
$$h^0(\mathcal{I}_X(d))=\max\left\{\binom{d+n}{n}-e(d+1), 0\right\}.$$
So for example take $X\subset \mathbb{P}^3$ to be the union of 5 skew lines in $\mathbb{P}^3$, and take $H\subset\mathbb{P}^3$ to be a generic hyperplane not containing any of those lines, and let $L$ be a nonzero linear form vanishing on $H$.  Then the defining ideal $I_H\subset\mathbb{C}[x_0,x_1,x_2,x_3]$ is the principal ideal generated by $L$, and in particular $I_H$ contains a 10-dimensional subspace of cubic forms.  Also taking $d=3$, $e=5$, $n=3$ in the theorem, we see that $I_X$ contains no cubic forms, and hence the sum $I_X+I_H$ contains a 10-dimensional subspace of cubic forms.  On the other hand, the intersection of $X$ and $H$ is $5$ points, and the defining ideal of $5$ points contains at most a 15-dimensional subspace of cubic forms (I'm just thinking of these naively as solutions to a $5\times 20$ homogeneous linear system).  Note that $X$ is not Cohen-Macaulay here.  This example is due to Juan Migliore.
