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Suppose I have a Cartier divisor D in a smooth variety X, and suppose I have a subvariety Y in X. Is it always possible to talk about restrict D to Y even when Y might be contained in D? I think it is true from a bundle point of view, because when I look at the example for standard blow up of Cn at the origin and restrict the exceptional divisor to itself, it makes sense. So my question is: what if I restrict D to a singular subvariety Y contained in D.

  1. Does it make sense. How to understand it?
  2. If the restriction is meaningful, is the restriction still Cartier in Y
  3. If D is only a Weil divisor, in general, is it meaningful to restrict it to a normal subvariety contained in D?

I really want to make it clear.

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    $\begingroup$ You essentially already answered your questions 1 and 2. For 3: if X is smooth, every Weil divisor is Cartier. If it isn't, then restriction doesn't always work. $\endgroup$ May 30, 2017 at 19:08
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    $\begingroup$ I would personally say that you cannot restrict the divisor, but you can restrict the divisor class, at least in the case of a Cartier divisor. For Weil divisors, even that might be out of reach. For an explicit example, see Example 11.4.6 in Fulton's Intersection Theory. $\endgroup$ May 30, 2017 at 22:30

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  1. Chow's Moving lemma says given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' on X such that Z' is rationally equivalent to Z and Y and Z' intersect properly. That means there exists a Weil divisor D' (since X is smooth Cartier, Weil, Line bundles all are equivalent) which is linearly equivalent to Weil divisor D and intersect Y along a codimension 1 sub variety of Y. This may not be a Weil divisor(Since Y may not be locally factorial etc.). Also this restriction here is not unique in any case (as R. van Dobben de Bruyn was saying).

  2. It is a Cartier divisor if you assume Y is Projective or Integral. Because for projective ( or Integral) varieties Cartier and line bundles are equivalent (Hartshorne Remark 6.14.1).

  3. same as 1.

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