show/hide this revision's text 4 added 1 characters in body

Dear Eugene,

I think I can answer one of your questions, namely about $Z$ having rational singularities. It does and here is why:

$Z$ is indeed normal by construction.

Since $f$ is a Mori-contraction, $-K_X$ is $f$-nef and $f$-big. (In fact, since the relative Picard number is $1$ it is $f$-ample). I claim that $R^if_*\mathcal O_X=0$ for all $i>0$. If $f$ were birational, this would be a simple consequence of Kawamata-Viehweg vanishing. In this case we need to work a little more.

First, by Kollár's torsion-freeness theorem (see [Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987.] for the format needed here and see the references there for proofs) these sheaves are torsion-free. (Not in general, just in this case!).

Second, let $U\subseteq Z$ be a non-empty subset over which $f$ is flat. The fibers over $U$ are Fano varieties and hence $H^i(F,\mathcal O_F)=0$ for $i>0$ for a fiber $F$. I probably should have taken a resolution to start with and then one would not have to worry about singularities, but at least for a general $F$ one can say that $(F,\Delta_{|F})$ is klt and hence Kawamata-Viehweg vanishing applies, so we get the above vanishing. Anyway, then by Grauert's theorem [Hartshorne, III.12] it follows that $(R^if_*\mathcal O_X)_{|U}=0$ for all $i>0$. However, we have already established that these sheaves are torsion-free, so if they are zero on a non-empty open set then they are zero everywhere.

OK, so we get that $R^if_*\mathcal O_X=0$ for all $i>0$. Now we are only almost there because the original definition of rational singularities would require this for a birational morphism and $f$ is decidedly not birational. But this still implies that $Z$ has rational singularities by SJK: A characterization of rational singularities.

Now, regarding whether $Z$ can be klt, canonical, etc. One simplification I can suggest is that you do not need the $\Delta'$ there. If you find a $\Delta'$ such that $(Z,\Delta)$ (Z,\Delta')$ is klt, dlt, lc, etc, then so is $Z$ since by the $\mathbb Q$-factoriality $\Delta'$ will be $\mathbb Q$-Cartier. See [Kollár-Mori, 2.35].

show/hide this revision's text 3 deleted 718 characters in body

Dear Eugene,

I think I can answer one of your questions, namely about $Z$ having rational singularities. It does and here is why:

$Z$ is indeed normal by construction.

Since $f$ is a Mori-contraction, $-K_X$ is $f$-nef and $f$-big. (In fact, since the relative Picard number is $1$ it is $f$-ample). I claim that $R^if_*\mathcal O_X=0$ for all $i>0$. If $f$ were birational, this would be a simple consequence of Kawamata-Viehweg vanishing. In this case we need to work a little more.

First, by Kollár's torsion-freeness theorem (see [Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987.] for the format needed here and see the references there for proofs) these sheaves are torsion-free. (Not in general, just in this case!).

Second, let $U\subseteq Z$ be a non-empty subset over which $f$ is flat. The fibers over $U$ are Fano varieties and hence $H^i(F,\mathcal O_F)=0$ for $i>0$ for a fiber $F$. I probably should have taken a resolution to start with and then one would not have to worry about singularities, but at least for a general $F$ one can say that $(F,\Delta_{|F})$ is klt and hence Kawamata-Viehweg vanishing applies, so we get the above vanishing. Anyway, then by Grauert's theorem [Hartshorne, III.12] it follows that $(R^if_*\mathcal O_X)_{|U}=0$ for all $i>0$. However, we have already established that these sheaves are torsion-free, so if they are zero on a non-empty open set then they are zero everywhere.

OK, so we get that $R^if_*\mathcal O_X=0$ for all $i>0$. Now we are only almost there because the original definition of rational singularities would require this for a birational morphism and $f$ is decidedly not birational. But this still implies that $Z$ has rational singularities by SJK: A characterization of rational singularities.

Now, regarding whether $Z$ can be klt, canonical, etc. One simplification I can suggest is that you do not need the $\Delta'$ there. If you find a $\Delta'$ such that $(Z,\Delta)$ is klt, dlt, lc, etc, then so is $Z$ since by the $\mathbb Q$-factoriality $\Delta'$ will be $\mathbb Q$-Cartier. See [Kollár-Mori, 2.35].

So, whether or not $Z$ is klt, dlt, lc, etc., I am not sure. My feeling is that it does not follow that it is any of those and at least it does not follow obviously (or at least I can't see how). Of course, if $Z$ turns out to be Gorenstein for some reason, then having rational singularities implies that it is canonical. You know that it is $\mathbb Q$-factorial and since it has rational singularities, it is also Cohen-Macaulay, so the only thing missing is that $\omega_X$ should be invertible to make it Gorenstein. However, the conditions that you know are not enough to guarantee that. There are rational $2$-dimensional singularities that are $\mathbb Q$-factorial and rational but not even log canonical.

show/hide this revision's text 2 typo corrected

Dear Eugene,

I think I can answer one of your questions, namely about $Z$ having rational singularities. It does and here is why:

$Z$ is indeed normal by construction.

Since $f$ is a Mori-contraction, $-K_X$ is $f$-nef and $f$-big. (In fact, since the relative Picard number is $1$ it is $f$-ample). I claim that $R^if_*\mathcal O_X=0$ for all $i>0$. If $f$ were birational, this would be a simple consequence of Kawamata-Viehweg vanishing. In this case we need to work a little more.

First, by Kollár's torsion-freeness theorem (see [Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987.] for the format needed here and see the references there for proofs) these sheaves are torsion-free. (Not in general, just in this case!).

Second, let $U\subseteq Z$ be a non-empty subset over which $f$ is flat. The fibers over $U$ are Fano varieties and hence $H^i(F,\mathcal O_F)=0$ for $i>0$ for a fiber $F$. I probably should have taken a resolution to start with and then one would not have to worry about singularities, but at least for a general $F$ one can say that $(F,\Delta_{|F})$ is klt and hence Kawamata-Viehweg vanishing applies, so we get the above vanishing. Anyway, then by Grauert's theorem [Hartshorne, III.12] it follows that $(R^if_*\mathcal O_X)_{|U}=0$ for all $i>0$. However, we have already established that these sheaves are torsion-free, so if they are zero on a non-empty open set then they are zero everywhere.

OK, so we get that $R^if_*\mathcal O_X=0$ for all $i>0$. Now we are only almost there because the original definition of rational singularities would require this for a birational morphism and $f$ is decidedly not birational. But this still implies that $Z$ has rational singularities by SJK: A characterization of rational singularities.

Now, regarding whether $Z$ can be klt, canonical, etc. One simplification I can suggest is that you do not need the $\Delta'$ there. If you find a $\Delta'$ such that $(Z,\Delta)$ is klt, dlt, lc, etc, then so is $Z$ since by the $\mathbb Q$-factoriality $\Delta'$ will be $\mathbb Q$-factorialQ$-Cartier. See [Kollár-Mori, 2.35].

So, whether or not $Z$ is klt, dlt, lc, etc., I am not sure. My feeling is that it does not follow that it is any of those and at least it does not follow obviously (or at least I can't see how). Of course, if $Z$ turns out to be Gorenstein for some reason, then having rational singularities implies that it is canonical. You know that it is $\mathbb Q$-factorial and since it has rational singularities, it is also Cohen-Macaulay, so the only thing missing is that $\omega_X$ should be invertible to make it Gorenstein. However, the conditions that you know are not enough to guarantee that. There are rational $2$-dimensional singularities that are $\mathbb Q$-factorial and rational but not even log canonical.

show/hide this revision's text 1