Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$. $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup R_k$ is a decomposition of $V$ into irreducible components. How can we characterize the set of points on $V$ that lies in at least two components? If this is hard to compute, is there a good approximation to this set (some bigger set that contains it)?

A second question related to the one above is: how can we find the equations that describe the singular loci (the difficulty lies in the fact that we don't know the dimension at the point we are interested in)? This loci will give an approximation to the set discussed above.


With regards to the first question, if the $R_i$ are all normal, then this is just the vanishing locus of the conductor ideal. This is not difficult to compute with something like Macaulay (via the method below). Without the normal hypothesis, the conductor gives a good approximation (which is better than the singular locus approximation you suggest in your question).

But the general question is easy too (and also easy to compute with Macaulay), consider the map $$R_1 \coprod R_2 \coprod \dots \coprod R_k \to V$$ corresponding to the inclusion $$f : O_V \subseteq O_{R_1} \oplus O_{R_2} \oplus \dots \oplus O_{R_k}.$$ Let $M$ be the cokernel of $f$ and simply compute $\text{Ann}_{\mathcal{O}_V} M$. The vanishing locus of this ideal will be exactly the points contained in at least two components.

EDIT: With regards to the singular locus question, compute the singular locus of each component individually and then union that set with the set computed above (which is also necessarily part of the singular locus).

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