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I have a number of vague questions that I wasn't sure whether they are suitable to ask or not, but I decided to ask!

According to this result, any two birational varieties can be constructed by a sequence of blow-ups and blow-downs from each other. (There are more conditions on the theorem but I am simplifying it). The way I see this theorem is that it makes each class of birational varieties very combinatorial in a sense. It means if you have access to one variety in a birational class you can constructed the rest of them by very specific types of operations.

  1. I was wondering whether there are a certain simple class of operations/surgeries that can make it possible to move from any variety to the other one (of fixed dimension and possibly only smooth varieties)?

  2. I was wondering if we add finite covers (not necessarily etale) to blow ups and blow-downs does it make moving between any two varieties possible? (I know finite covers are not quite surgeries but that is what came to my mind!)(The answer to this is positive but I don't think it implies anything regarding question 3.)

  3. Assume a certain statement has this property that if it is true for a finite cover then it is true for the base. Also it is a birational invariant. Is there any smallest family of specific varieties that you need only to prove the statement for that family in order to prove it for all varieties of the same dimension?

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  • $\begingroup$ Any projective variety of dimension $d$ is a finite cover of $\mathbb{P}^d$: just embed the variety in one big projective space and keep projecting away from points. So the answer to your question 2. is yes, if I interpreted it correctly. $\endgroup$
    – Daniele A
    Commented Feb 18, 2021 at 1:01
  • $\begingroup$ Yes you are right. This is basically Noether's normalization. $\endgroup$
    – user127776
    Commented Feb 18, 2021 at 1:05
  • $\begingroup$ Re question 3: The answer should be no for any formalization of this question (basically because by passing to finite covers you generally cannot make the variety less complicated). For instance, for every family of varieties, a very general hypersurface of large degree will admit no birational morphisms from any member of that family. $\endgroup$
    – dhy
    Commented Feb 18, 2021 at 4:36
  • $\begingroup$ @dhy I don't quite understand your argument. The family doesn't need to be bounded. For example you can take it to be product of curves. Now why no covering of a hypersurface of large degree can be birational to a product of curves? $\endgroup$
    – user127776
    Commented Feb 18, 2021 at 5:06
  • $\begingroup$ @user127776 It is true that my argument does not apply to that case. More generally though, statements of the form "every variety has a nice finite cover" tend to be false, which is unfortunate because they would be extremely useful. P.S. the case of products of curves has some interesting history behind it: see mathoverflow.net/questions/98771/… $\endgroup$
    – dhy
    Commented Feb 18, 2021 at 5:26

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