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Let $f:X\to Y$ be a degree $d$ morphism of complex projective varieties, and let $V\subset Y$ an irreducible subvariety, $W$ its preimage under $f$. I want to find all of the components of $W$.

Suppose that I've already found several components, $W_1,\ldots,W_k$, and the components that are known are such that the sum of the degrees of the maps $f|_{W_i}$ adds up to $d$, and I know that there exists at least one component where the map restricts to one that isn't finite.

How can I determine if there are any other components of this nature, that don't contribute to the degree of $f$? And if they exist, is there a good way to identify what they are?

(Note: this is an attempt to redo a question I asked a few days ago and deleted, hopefully, this is better phrased. Roughly, I'm looking for ways to find all the components of the preimage of a variety through a morphism as described above)

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  • $\begingroup$ Are you trying to locate the components of W for which the restriction of f is not dominant? $\endgroup$
    – damiano
    Commented Jun 25, 2010 at 19:56
  • $\begingroup$ The components are of varying dimension, I'm trying to locate the components of $W$ for which $f$ isn't finite, in particular, a strategy for determining that there are no more. $\endgroup$
    – Charles Siegel
    Commented Jun 25, 2010 at 20:02
  • $\begingroup$ I should say, finite onto its image, as the dimensions of the components vary $\endgroup$
    – Charles Siegel
    Commented Jun 25, 2010 at 20:02
  • $\begingroup$ I suspect you need to know more about your problem: you could always precompose f with the blow up in X of any number of divisors on W and the degree would not change, but the number of irreducible components of the new W would. For instance, if X=Y is a surface and V is a curve, you can replace id: X-->Y by the blow up of r points on V and you would get r+1 components in the inverse image of V in X. $\endgroup$
    – damiano
    Commented Jun 25, 2010 at 20:12
  • $\begingroup$ In any case, there is a closed subscheme Z in Y such that the fiber dimension of f above points in Z is bigger than zero, and it seems that your assumptions imply that V is not contained in Z. The components of the intersection of V and Z that have codimension one in V will certainly give rise to components in W for which the restriction of f is not finite. $\endgroup$
    – damiano
    Commented Jun 25, 2010 at 20:36

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