# Questions tagged [minimal-model-program]

minimal model program is part of the birational classification of algebraic varieties.

114
questions

1
vote

0
answers

102
views

### Modifying the base of a rational map

Let $f : X \dashrightarrow S$ be a rational map of smooth projective varieties. Is it true that, after a birational modification of $S$, every fiber intersects the domain of definition? Explicitly, is ...

4
votes

2
answers

139
views

### Mori cones and projective morphisms

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely ...

1
vote

0
answers

41
views

### Singularities of toric pairs

Suppose $(X,B)$ is a log canonical pair and $f: X \rightarrow Y$ an equidimensional toroidal contraction such that every component of $B$ is $f$ -horizontal. Let $\Gamma$ denote the reduced ...

1
vote

1
answer

162
views

### Two morphisms possess the same Viehweg's variation

Recall the definition of Viehweg's variation intimated from Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces,
E. Viehweg
Let $f: V\rightarrow W$ be a fiber space (...

1
vote

0
answers

101
views

### Numerical reduction map for line bundles?

For a nef line bundle $L$ on a normal projective variety $X$, we have three invariants- the nef dimension $n(L)$, the numerical dimension $\nu(L)$ and the Iitaka dimension $\kappa(L)$. $n(L)$ is ...

4
votes

1
answer

124
views

### Isomorphism outside of negative curves against the canonical

Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are non-positive against the canonical divisor is a closed subset $F\subsetneq ...

2
votes

1
answer

143
views

### Singularities of contractions of extremal faces

Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K_X + \Delta)$-...

2
votes

0
answers

162
views

### Description of movable cone

Let $X$ be a normal, $\mathbb{Q}$-factorial projective variety (over $\mathbb{C}$). If we assume that $X$ is a Mori dream space, then by definition its movable cone is rational polyhedral, and there ...

7
votes

1
answer

476
views

### Is there a classification of minimal algebraic threefolds?

The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what ...

7
votes

0
answers

329
views

### Where does the word "log" in log pair come from?

The minimal model program works with pairs $(X,B)$ where $X$ is a variety and $B$ is a certain kind of divisor on it. I've seen these described as "logarithmic pairs". There are also "...

1
vote

1
answer

138
views

### Positivity of the global log canonical threshold of a pair

Let $(X,L)$ be a polarized smooth projective variety. Let $D$ be a smooth irreducible divisor in $X$. Let $0<c<1$ be a real number. We denote $cD$ as $\Delta$. We can define the $\alpha$ ...

3
votes

1
answer

154
views

### Existence of terminal $3$-fold flips

Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...

4
votes

0
answers

120
views

### Parameter spaces for conic bundles

A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...

3
votes

0
answers

187
views

### Mori cone of Picard rank two varieties

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...

1
vote

0
answers

60
views

### On the b-nefness of the moduli part of canonical bundle formula

I have recently been wondering about the existence of a canonical bundle formula in the following situation and am not sure how to proceed.
Suppose $(X,B) \xrightarrow{f} Y$ is a fibration where $ (X,...

1
vote

0
answers

73
views

### Canonical covering stack of a flop

In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...

2
votes

1
answer

150
views

### Restriction of small transformations

Let $\phi:X\dashrightarrow Y$ be an elementary small transformation (isomorphism in codimension $1$) between normal and $\mathbb{Q}$-factorial projective varieties.
Then there are small contractions $...

2
votes

0
answers

198
views

### descent of nef divisors

Let $f:X \rightarrow Y$ be a flat surjective morphism of complex projective varieties with connected fibers, $X$ normal, $Y$ smooth and $dim (Y) < dim(X)$. Suppose $L \in Pic (X)$ such that $L|_{X_{...

2
votes

0
answers

98
views

### Cone and contraction theorems for certain sub-klt pairs

Suppose $(Y,B_Y)$ is a sub-klt pair which is a birational model of a klt pair $(X,B)$: there exists a birational morphism $\pi:(Y,B_Y) \rightarrow (X,B)$ such that $\pi^{*}(K_X+B)=K_Y+B_Y$. We know ...

4
votes

1
answer

333
views

### Termination of a minimal model program

I am reading "The dual complex of
singularities" by de Fernex, Kollár
and Xu and in the proof of Corollary 24 I have encountered a bit of
reasoning that confuses me.
Let $(X, \Delta)$ be a $\...

1
vote

1
answer

280
views

### Log resolution of a variety of log general type

Work over the complex numbers. Let $(B, \Delta)$ be a normal irreducible variety of log general type, i.e., $K_B + \Delta$ is ample. Let $f : (\widetilde{B}, \widetilde{\Delta}) \to (B, \Delta)$ be a ...

1
vote

0
answers

361
views

### Minimal model vs canonical model of a surface

When I have a projective surface $X$, for simplicity smooth, I can find a simpler smooth surface on its binational class. In this way we find in a finite number of steps the simplest surface $Y$, i.e. ...

5
votes

0
answers

152
views

### Steps of the MMP "in family"

Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...

1
vote

1
answer

276
views

### Prescribing the discriminant locus of fiber spaces

Let $X$ be a projective manifold with $\dim_{\mathbb{C}} X \geq 3$. Assume $X$ is the total space of a fiber space, i.e., there is a proper surjective holomorphic map $f : X \to Y$ with connected ...

1
vote

0
answers

80
views

### How to show a contraction of singular moduli space is projective?

Let $\mathcal{H}$ be a certain kind of Hilbert scheme of curves on some smooth projective variety $X$ and $\mathcal{H}$ is projective and irreducible of dimension $3$. There is a divisor $\mathcal{D}\...

1
vote

0
answers

211
views

### Does nefness carry over through flips?

Suppose $\pi: X \dashrightarrow Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We ...

1
vote

1
answer

174
views

### Terminal $\mathbb{Q}$-factorial divisorial contractions

Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$....

0
votes

1
answer

244
views

### On the birational equivalent class of algebraic surfaces with Picard number $1$

An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$.
Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...

2
votes

1
answer

336
views

### Derived category of singular varieties

Let $X$ be a projective variety with only normal crossing singularity. Is there a description of the derived category or the category of perfect complexes? What about the existence of semiorthogonal ...

5
votes

1
answer

461
views

### Relative logarithmic cotangent bundle

Let $\mathcal X \rightarrow S$ be a flat family of projective varieties over a discrete valuation ring $S$ such that the generic fibre $\mathcal X_{\eta}$ (say) is smooth projective variety and the ...

1
vote

1
answer

421
views

### Are terminal singularities $ \mathbb{Q}$-factorial?

The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need ...

2
votes

0
answers

73
views

### Finding divisors with canonical singularities in a moving linear system

I apologize if the question is too naive or trivial:
We know that any reduced divisor in a smooth variety has Gorenstein singularities. However, I don't know if there's a cone theorem for Gorenstein ...

2
votes

2
answers

407
views

### One point compactification of the tangent bundle

Is there a smooth variety $X$ which is a one point compactification of the tangent bundle of $\mathbb P^1$?

22
votes

1
answer

804
views

### Is being of general type stable under generization

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.
Definition. An integral projective ...

3
votes

0
answers

129
views

### Is the generalized Kummer threefold rational in characteristics 3?

Let $E_i\!: y_i^2 = x_i^3 - x_i$, $i = 1, 2, 3$ be three copies of the supersingular elliptic curve in characteristics $3$. Consider on $E_i$ the following automorphism of order $3$:
$$
\sigma(x_i,...

5
votes

1
answer

274
views

### $K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?

Let $(X,B)$ be a projective log canonical pair (here I mean $B \geq 0$). Assume that the coefficients of $B$ are rational, and that $K_X+B \equiv 0$. Is it true that $K_X + B \sim_\mathbb{Q} 0$? I ...

1
vote

1
answer

183
views

### Small contraction for Hyperkähler Varieties

I have the following basic question. Everything is over $\mathbb{C}$.
Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow ...

2
votes

2
answers

416
views

### Intuition behind Kawamata's definition of a relative movable Cartier divisor

I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]:
[1] Y. Kawamata, ...

2
votes

1
answer

343
views

### Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...

3
votes

1
answer

260
views

### Movable divisor with base locus on a hyperkahler variety

I'm looking for an example of the following:
$X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$);
$D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of
$D$ is a ...

2
votes

0
answers

464
views

### Small contractions as blow ups

To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.
I am trying to learn a bit about birational morphisms:...

3
votes

1
answer

453
views

### How to split a Multi-section into finitely many Sections via base-change?

Let $:f:X\to Y$ be a projective surjective morphism between two normal varieties over $\mathbb{C}$. Assume that $f$ has only $1$-dimensional fibers. Let $D$ be a multi-section of $f$, i.e., $D$ is a ...

2
votes

1
answer

192
views

### Flatness of Fano Contractions

In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the ...

7
votes

1
answer

392
views

### Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...

1
vote

0
answers

134
views

### Compactifying morphisms and ample line bundles

Let $f:X\to Y$ be a projective morphism between two normal quasi-projective varieties, and $L$ a $f$-ample line bundle on $Y$. Then the claim is: There is a compactication $\bar{f}:\overline{X}\to\...

1
vote

0
answers

49
views

### How can I describe in explicit geometric terms the (in general non-complete) linear system?

Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...

1
vote

0
answers

218
views

### Section ring $R(X,D)$ of $D$ is finitely generated if $\kappa (X,D) \leq 1$

In the remark on the bottom of page 5 of this paper, the author states that
It is well-known fact that the algebra of a Divisor $D$ with $\kappa (X,D) \leq 1$ is finitely generated over $k$. In ...

5
votes

0
answers

213
views

### Map associated to linear system onto curve is morphism

In Mumford's first paper on Surfaces in char $p$ [1], part 2 Step (II), he wants to show that, given an indecomposable curve of canonical type $D$ on a smooth projective surface $F$ with $p_g(F)=0, ...

5
votes

0
answers

113
views

### Minimal Model Program for sub-lc pairs

In many articles of the minimal model program the authors work with sub-lc pairs instead of lc-pairs. In other words, they consider non-necesarilly effective boundary divisors $B$.
Is it expected (...

6
votes

0
answers

509
views

### Pseudo-effective divisor which is not nef in any birational model

Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef
if there exists a birational ...