- (Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. The question is, can we detect if $Y$ is reduced as a subscheme on generic points of $Y$? More precisely, assume there is a (proper) closed subvariety $Z$ of $Y$ such that for all $y \in Y \setminus Z$, the localization of $I$ at the maximal ideal of $y$ is a radical ideal. Then is it true that $I$ is a radical ideal?

The answer to the naive question is false, even in the simplest case that $I$ is principal, e.g. take $X = Spec~ k[x,xy,y^2]$ and $I$ to be the ideal generated by $x$. Then $I$ is radical on $X\setminus\{O\}$, where $O$ is the "origin". A perhaps more natural set of counter-example can be constructed as follows: Take a subvariety $Y'$ of $X$ which is not a complete intersection, take a minimal collection of generators $g_1, \ldots, g_k$ of the ideal of $Y'$, and set $I$ to be the ideal generated by $g_1, \ldots, g_{k-1}, g_k^2$. So the `real question', which is perhaps too broad, is:

- Are there 'natural' conditions on $X$ and $I$ under which the reducedness of $V(I)$ can be detected at generic (closed) points on $Y$?

A 'natural' situation that I have in mind (and that disallows both counter-examples above) is the following:

- Is the answer to Question 2 affirmative when $X$ is normal and $I$ is generated by a regular sequence, i.e. $Y$ is a set-theoretic complete intersection? (The answer is affirmative in the simplest case, i.e. when $I$ is principal.)