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Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ does not have any component that map to $P$?

More generally, if $W$ is a proper subvariety of $V$, under what conditions there is no component of $E$ that maps to $W$?

In my case I have a fixed subvariety $V$ (which can be arbitrarily singular), a fixed point $P \in V$, and an ideal $I$ whose zero set is $V$. But I have some freedom in choosing this $I$. So what would be perfect is a condition that can be checked say on a set of generators of $I$ and the ideal of $P$.

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  • $\begingroup$ I'm not sure about the term algorithm here. Certainly blowing up is algorithmic, I'm sure I could write a Macaulay2 function say that checked this for a fixed $I$ and $P$. Indeed, this probably already mostly exists via functions already implemented around reesAlgebra. It sounds like what you really want though is to show that for some sufficiently carefully chosen $I$, you can avoid any component mapping to a point $P \in V$. I would guess that this is probably possible (assuming $\overline{\{ P \}} \neq V$) but I don't have any idea how to do it... $\endgroup$
    – Karl Schwede
    Commented Feb 24, 2015 at 4:23
  • $\begingroup$ @KarlSchwede: yes, $\overline{\lbrace P \rbrace} \neq V$, and you are right in all other counts. $\endgroup$
    – pinaki
    Commented Feb 24, 2015 at 4:38

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