The minimal-model-program tag has no wiki summary.

**1**

vote

**1**answer

184 views

### On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...

**2**

votes

**1**answer

161 views

### Finite generation of certain $\mathcal{O}_X$-algebra

It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the ...

**1**

vote

**1**answer

132 views

### Run MMP between varieties of isomorphic in codimension 1

Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with ...

**2**

votes

**1**answer

183 views

### A question about running MMP with scaling

Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective.
Suppose $K_X + \Delta$ is not nef (over $U$) and there ...

**2**

votes

**1**answer

196 views

### Number of minimal models of a surface

I would like to know if the following statement is true or false:
Given a non-singular complex projective surface $S$, it has at most a countable number of minimal models (up to isomorphism).
...

**0**

votes

**2**answers

191 views

### Decompose a big divisor as nef big divisor and effective divisor

Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a ...

**7**

votes

**0**answers

308 views

### Singularities arising from the Minimal Model Program (an algebraic point of view)

I will start the story by the end:
Is there some characterization of (some of) the singularities arising from the Minimal Model Program (canonical, terminal, log-...) in terms of commutative algebra ...

**7**

votes

**1**answer

233 views

### Is the number of minimal models finite

Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...

**0**

votes

**1**answer

195 views

### A covering lemma of Kawamata

In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is
Let $X$ be a non-singular projective variety, and $D$ be a divisor ...

**1**

vote

**1**answer

195 views

### Properties of extreme rays

Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that ...

**0**

votes

**1**answer

225 views

### Kawamata-Log-Terminal pairs

Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$.
Consider the ...

**2**

votes

**0**answers

278 views

### Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...

**1**

vote

**1**answer

153 views

### On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier

Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( ...

**5**

votes

**1**answer

275 views

### Is there an Enriques–Kodaira-like classification of Fano threefolds?

I am mainly interested in varieties over an algebraic closed field $k$ (or $\mathbb{C}$). The classification of complex surface is established in the last century and known as Enriques–Kodaira ...

**2**

votes

**1**answer

303 views

### Minimal semistable model for K3-surfaces.

I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...

**6**

votes

**2**answers

1k views

### Training towards research on birational geometry/minimal model program

Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having ...

**1**

vote

**1**answer

695 views

### Trivial canonical bundle

Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or ...

**2**

votes

**0**answers

147 views

### Controlling singularities on log mmp

Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.
If I do (relative) ...

**5**

votes

**1**answer

741 views

### Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...

**6**

votes

**0**answers

474 views

### Are conical symplectic resolutions Mori dream spaces?

This is one of these questions where it's tempting to just leave it at the title, but let me try to define the objects in question.
A conical symplectic resolution is a projective resolution of ...

**3**

votes

**1**answer

376 views

### References about pseudoeffective cone

I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.

**9**

votes

**1**answer

483 views

### Infinitely many minimal models

There are examples of elliptic fiber spaces over a two-dimensional base which have infinitely many relative minimal models (where two abstractly isomorphic models connected by flops are counted ...

**1**

vote

**1**answer

370 views

### Why are the different definitions of minimal model equivalent?

I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:
$K_X$ is nef.
Every birational morphism from $X$ ...

**8**

votes

**2**answers

811 views

### The minimal model program and symplectic resolutions

I've been reading some papers of Namikawa lately, and have on several occasions come across a claim I would really like someone to expand on.
On page 4 of Poisson deformations of affine symplectic ...

**5**

votes

**3**answers

582 views

### Basepoints in the Canonical System of Algebraic Surfaces

Let X be an smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...

**8**

votes

**3**answers

1k views

### Singularities of pairs

In the next days I have to give a talk in which I need to explain some of the usual singularities of pairs that one meets when dealing with the minimal model program: KLT, DLT and LC pairs.
In ...

**4**

votes

**2**answers

616 views

### Possible singularities of the base of a Mori fiber space

Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the ...

**7**

votes

**2**answers

763 views

### How much can small modifications change the nef cone?

First let me give a precise formulation of the question; I'll give some background/motivation at the end.
If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently ...

**11**

votes

**3**answers

2k views

### Does negative Kodaira dimension imply uniruled?

There is a conjecture (often attributed to Mumford) I believe which states that if, on a smooth proper variety $X$ (over an algebraically closed field of characteristic zero), there are no ...

**11**

votes

**2**answers

1k views

### What is known about the MMP over non-algebraically closed fields

I would like to know how much of the recent results on the MMP (due to Hacon, McKernan, Birkar, Cascini, Siu,...) which are usually only stated for varieties over the complex numbers, extend to ...

**18**

votes

**5**answers

2k views

### Flips in the Minimal Model Program

In order get a minimal model for a given a variety $X$, we can carry out a sequence of contractions $X\rightarrow X_1\ldots \rightarrow X_n$ in such a way that that every map contracts some curves on ...