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Suppose I have a singular projective variety defined by some homogeneous equations in complex projective space. Is the resolution of singularities effective? That is, do I actually know which smooth centers to blow up?

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Yes, in the sense that resolution of singularities is implemented in the computer algebra package Singular. See the manual of Singular for references. (There might be other/better references.) However, if I remember correctly the centers are not unique.

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    $\begingroup$ There are ways to choose the centers canonically. This is called "canonical resolution of singularities", see e.g. Bierstone, Edward; Milman, Pierre D. (1997), "Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant.", Invent. Math. 128 (2): 207–302. $\endgroup$
    – Sasha
    Commented Aug 25, 2010 at 8:10
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    $\begingroup$ I just want to emphasize that there are many "canonical" ways to choose the centers. $\endgroup$
    – Karl Schwede
    Commented Aug 25, 2010 at 16:58
  • $\begingroup$ and that effective can still mean a lot of iterations logic.pdmi.ras.ru/~grigorev/pub/hiron-complex_journal.pdf $\endgroup$
    – O.R.
    Commented Jun 3, 2011 at 0:59

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