# restriction of a divisor

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.

• 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. May 30, 2017 at 19:08
• 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. May 30, 2017 at 22:30