4
$\begingroup$

Every variety here is over $\mathbb{C}$. Let $f: X \rightarrow C $ be a flat proper surjective morphism from a quasi-projective variety $X$ to a quasi-projective smooth curve. Let $p\in C$. We assume that the underlying variety $X_{p}=f^{-1}(p)$ is normal, and has $Q$-Gorenstein log-terminal singularities (or Gorenstein canonical singularities), and $f: X^{*}=X-X_{p} \rightarrow C-p=C^{*}$ is smooth. However, $X_{p}$ may not be reduced. We further assume that relative canonical bundle $K_{X^{*}/ C^{*}}$ is trivial.

By Mumford semi-stable reduction theorem, we have a new family $f': X'\rightarrow C'$ such that $X'$ is smooth, $p\in C'$, $X'_{p}$ has simple normal crossing singularities, and $X'-X'_{p} \rightarrow C'-\{p\}$ is a finite cover.

Now we run the minimal model program. By using the result in for example Theorem 1.1 of this paper http://arxiv.org/abs/1010.2577, we have the third family $f'' : X'' \rightarrow C' $ such that the canonical divisor $K_{X''}\sim_{C'} 0$. Note that $X''$ is birational to $X'$, and any fiber $X"_{q}$, $q\neq p$, comes from the original family $X$. We can view these three special fibers as some sort of different filling in.

Question: What is the relationship between the special fiber $X_{p}$ in the first family and $X''_{p}$ in the third family, if they have some at all? Is that possible after taking some flops of $X''$, $X''_{p}$ is some kind of covering space of $X_{p}$?

I'm not an algebraic geometer, I hope the question makes sense. Thanks.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.