User cx - MathOverflow

Answer by CX for blow-ups and singularities

2013-05-24T22:57:47Z

I think this is true. Taking a point $x\in X$, and base change your setting up to its formal neighborhood, we can assume $K(X)\subset K(Y)$ is a Galois extension with finite abelian group.

Now we can put a root stack structure $\mathcal{X}\to X$ which is branched over $D$ such that the morphism $Y\setminus\pi^{-1}(V)\to \mathcal{X}\setminus{V}$ is etale where $V\subset \mathcal{X}$ is of codimension at least 2 (See Matsuki-Olsson's paper). Then by purity, this indeed gives a DM-stack $\mathcal{Y}$ which is finite etale over $\mathcal{X}$. In particular $\mathcal{Y}$ is smooth and it has Y as its coarse moduli space. (By the way, I think once you assume $Y\to X$ branches over a snc divisor and $Y$ is normal, it comes for free that $Y$ has only abelian quotient singularities.)

Now the condition you have just implies the preimage $\mathcal{Z}$ of $Z$ in $\mathcal{X}$ is smooth. So its preimage $\mathcal{Z}_{\mathcal{Y}}$ in $\mathcal{Y}$ is smooth.

But then $Bl_{\mathcal{Z}_{\mathcal{Y}}}\mathcal{Y}$ is smooth, so its coarse moduli space only has abelian quotient singularities, which is precisely $\tilde{Y}$.

Answer by CX for Minimal semistable model for K3-surfaces.

2013-05-22T22:46:46Z

The answer is yes when p>3. Look at Kawamata's paper

Semistable minimal models of threefolds in positive or mixed characteristic. J. Algebraic Geom. 3 (1994), no. 3, 463–491.

and a correction in

Index 1 covers of log terminal surface singularities. J. Algebraic Geom. 8 (1999), no. 3, 519–527.

Lifting vector fields to its resolution in char $p$

2013-03-09T14:11:59Z

In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log resolution $\tilde{X}$, with logarithmic poles along the exceptional locus. Roughly speaking, the lifting of the tangent vector field automatically doesn't have deep poles.

Unfortunately, the proof there was only in characteristic 0. My question is

If we assume that resolution of singularity (any reasonable version you want) holds in char $p$, can we prove such a result in char $p$ as well?

Answer by CX for Do finitely many plurigenera determine the Kodaira dimension?

2013-03-09T13:52:28Z

I guess in general the answer is NO.

For instance if you take $X$ to be an $n$ dimensional variety which is $Y\times E$, where $E$ is an elliptic curve and $Y$ is an $n-1$ dimensional variety of general type with fast growing pluricanonical general, say $Y$ is a hypersurface of large degree $d$, then for any fixed $M$, if you choose $d$ sufficiently large, $h^0(X,iK_X)$ can beat any sequence of numbers $a_i$ $(0\le i\le M)$ which you wrote for a fixed $n$-dimensional general type variety.

Answer by CX for Open problems in Birational Geometry, after BCHM

2012-12-02T01:53:40Z

Let me add my answer.

In characteristic 0.

Abundance conjecture. This is probably universally accepted as the most important question for the current minimal model program after BHCM's proof of finite generation. I want to remark that n-dimensional log canonical MMP follows from n-dimensional klt MMP and n-1 dimensional log canonical MMP. So there is no new difficulty for log canonical pairs at least for MMP other than abundance for klt.
General type. In fact, after BCHM, Koll\'ar and many other people's work, I think we have a pretty good understanding of the rough classification of general types, namely, we know they form a moduli space which has a pretty reasonable compactification.
Fano and singularities. The boundedness of Fano, namely the BAB-conjecture is one of the most fundamental question about singular Fano. We only know special cases. In philosophy, there is a local-to-global principle, and correspondingly, we should consider certain boundedness of singularities. The famous challenge there is Shokurov's ACC of mld conjecture. Note this should be substantially harder than ACC of log canonical thresholds which now is a theorem. Another one is the semi-continuity of mld conjecture. And Shokurov proved these two conjectures together imply termination of flips.
Calabi-Yau. There are two fundamental problems there to me. One is the finiteness of the topological type, Another one is Kawamata-Morrison Cone conjecture. Each of them is still out of reach even in dimension 3. This is a not a big surprise, since we are still quite lack of understanding of CY in dimension 3. On the other hand, if you apply local-to-global principle here, then we should consider semi-log-canonical CY. And Koll\'ar's recent examples point out that the slc picture is even more complicated.

In characteristic p.

In char p, the most important question to me is the resolution of singularity. The next thing is how many results in characteristic 0 MMP still hold in characteristic p. WIthout vanishing type theorem, we can still formulate most of those results, but then it really puts a question mark on whether we should still believe them.

My feeling is although there are many substantial work there recently, it's still not clear what the picture should be.

Bertini's theorem in char p for base point free linear system

2012-01-20T02:13:34Z

I always believed the following statement: if $X$ is a smooth variety over an algebraically closed field of positive characteristic, assuming we know that the general member of a base point free linear system $|L|$ is reduced, then indeed a general member is smooth.

However, I realize this is not obvious, though all the examples I know which fail this Bertini theorem has non-reduced fibers.

So I was wondering whether indeed this statement is true. Or counterexamples are known.

[REEDIT:] After Laurent Moret-Bailly's nice counterexample. I was then wondering whether a counterexample exists when the linear system induces a birational morphism to its image. The assumption now is closer to the one in Bertini's theorem which says the statement is true when we assume the morphism is an embedding.

This sort of counterexamples will always help birational geometers to think what's going on in characteristic $p$.

Answer by CX for Numerically negative exceptional divisor on a surface.

2012-01-19T20:57:09Z

See the proof ON ISOLATED RATIONAL SINGULARITIES OF SURFACES (Artin) Propostiion 2 (i).

Answer by CX for Minimal Model Program for surfaces over algebraically closed fields of characteristic p

2012-01-11T17:44:22Z

In the surface case, MMP in char p is known. See Koll'ar-Kov'ac's preprint on Koll'ar's webpage.

In dimensional 3, the existence of divisorial contractions and flipping contractions is known as EWM (so the target is only known as a algebraic space). See Keel's paper BASEPOINT FREENESS FOR NEF AND BIG LINE BUNDLES. I'm not sure about the termination of flips. The existence of flips is certainly not known (at this moment).

In higher dimensions, I think almost nothing is known.

Answer by CX for Does an essential resolution of 2-dimensional hypersurface singularity preserves

2012-01-14T00:57:52Z

The answer is surely no.

If $D$ itself does not have a minimal log smooth resolution, then certainly $(V,D)$ couldn't have such a log resolution you need. On the other hand, there are bunch of isolated surface singularities whose minimal resolution is not log smooth, e.g., the log canonical surface singularity whose minimal log resolution has its exceptional locus an nodal rational curve.

Answer by CX for The minimal model program and symplectic resolutions

2011-04-15T23:36:04Z

The answer to the first question: if $X$ has klt singularities, then there exists a $\mathbb{Q}$-factorial variety $Y$ with a birational morphism $\pi: Y\to X$, such that if you write $\pi^*K_X=K_Y+\Delta$, then $\Delta$ is effective, and for any exceptional divisor $E$ of $Y$, we have the discrepancy $a(E,Y,\Delta)>0$, i.e., $(Y,\Delta)$ is terminal. This implies $Y$ itself is terminal. If you start with $X$ with only canonical singularities (This is stronger than klt singularities. But if $K_X$ is Cartier and $X$ is klt, then $X$ has canonical singularities. I mention this since I know in some cases from the representaion theory, indeed $K_X$ is trivial.), then $\Delta=0$. This follows from Corollary 1.4.3 of [BCHM]. In fact, it was known before that certain part of MMP would imply the existence of terminalization. The cases of MMP established in [BCHM] contain this part.

Answer by CX for Possible singularities of the base of a Mori fiber space

2010-10-15T15:48:03Z

Under the assumption that $X$ is $\mathbb{Q}$-factorial, section 5 of the paper http://arxiv.org/pdf/math/0606666 addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the pair $(X,\Delta)$ is klt, so is the base $(Z,\Delta_Z)$ for some $\Delta_Z$. As the example of Prokhorov shows, this is optimal, namely, the base may not be terminal (canonical) even you assume the $(X,\Delta)$ is. For lc case, I think dlt modification+perturbation reduce the question to the klt case.

Comment by CX

2013-05-24T00:24:36Z

What do you mean by canonical models? A family of K3 has relative trivial canonical class.

Comment by CX

2013-03-09T14:56:28Z

As the above post said, the negative &-\kappa& case is implied by the effective Iitaka fibration conjecture， which I think is expected to be true but still widely open in general.

Comment by CX

2013-03-09T13:13:33Z

I'm a bit confused. In the second example you give, is it also a smoothing by itself as the quotient will identify $f_t$ and $f_{-t}$ as one smooth fiber for $t\neq 0$?

Comment by CX

2012-12-13T19:57:45Z

If you blow up the original surface S, you will always have curves with negative self-intersections. The question is asking for BIRATIONAL!

Comment by CX

2012-01-31T01:32:20Z

I know this book which is not currently in our library though. But I'm not sure kind of examples exist there, do they?

Comment by CX

2012-01-20T14:55:32Z

Dear Laurent Moret-Bailly: Thank you very much for this nice example! I was also wondering whether we can move one more step to ask when the linear system induces a birational morphism, whether the counterexample still exists.

Comment by CX

2012-01-20T04:53:54Z

To S\'andor: Thanks for your remark! Yes, my question is whether such examples exist.

Comment by CX

2012-01-17T22:19:19Z

To Garcia: What I said is all for the general log pair case. I reedit my answer because I realize that Keel only show in dimension 3, flipping and divisorial contractions exist as EWM. In other words, so far we only know the target space for these contracts exists as an algebraic space.

Comment by CX

2012-01-16T00:58:16Z

I see. You are right. We also need to consider the curves of discrepancy 0 which doesn't necessarily appear in the minimal resolution of the surface itself.

Comment by CX

2012-01-11T18:55:07Z

Dear Jason: By 'MMP', I mean precisely the things we need for running the minimal model program (in the surface case, it just means the contraction exists.). With further digging on the literature, it might be also true that the abundance is also known for the surfaces pairs in char p. On the other hand, in the paper "Strong Rational Connectedness of Surfaces", there is an example due to Koll'ar giving a del Pezzo surface with only A_1 singularities, whose smooth locus doesn't contain free curves. Of course it still may be rationally connected, which I didn't really check.

Comment by CX

2010-10-15T17:32:08Z

To Sandor: surely we can forget $\Delta_Z$. But I write $\Delta_Z$ because I want to emphasize in the proof the discriminant divisor will appear, and for many questions we should take it into account.