Let X be a seminormal variety and S be the singular locus of X. Is Blow$\_S X=$ the normalisation of X?


Is the singular locus given by the conductor ideal?



singular locus of semi-normal variety
First, definitely the singular locus is not equal to the conductor. If $X$ is normal, the singular locus of $X$ is definitely not given by the conductor (which is the unit ideal and doesn't vanish anywhere).


For the first question however, Greco-Traverso actually studied blowing up the *conductor* in a seminormal variety $X$. For example, they showed the following.


**Theorem** *Suppose $X$ is an S2 seminormal scheme and $B$ is the conductor. If the normalization of $X$ is factorial (for example regular), then the normalization is equal to the blowup of the conductor.*


See section 7 of Greco-Traverso's *On seminormal schemes* for this and some slightly more general results. They also have some references to some other papers where blowing up the conductor is considered, for example Wilson, *On blowing up conductor ideals*


