Consider a smooth algebraic variety $X$ over $\mathbb C$. Suppose that one has two smooth closed subvarieties $Y_1$ and $Y_2$ and let $Z = Y_1\cap Y_2$. Denote by $I_1$ the ideal corresponding to $Y_1$ and by $I_2$ the ideal corresponding to $Y_2$. It is not hard to prove that if $Y_1$ intersects $Y_2$ transversally at any point then $\sqrt{I_1+I_2} = I_1+ I_2$. On the opposite site, if the locus of points of $Z$ where the intersection is not transversal has codimension $1$ in Z, then $\sqrt{I_1+I_2}$ is not equal to $I_1+ I_2$ in general.
The question: is it true that $\sqrt{I_1+I_2} = I_1+ I_2$ if locus $\{p\in Z\ \mid\ \text{intersection at $p$ is not transversal}\}$ has codimenstion at least $2$ in $Z$?
