# Questions tagged [minimal-model-program]

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

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### The embedding dimension and multiplicity of this singularity

Let $k$ be a algebraically closed field of characteristic 2, $Y_0:=G_m^3=\operatorname{Spec}k[x,x^{-1},y,y^{-1},z,z^{-1}]$
a three dimensional algebraic torus, $X=Y_0/(\mathbb{Z}/2\mathbb{Z})$ a ...

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### About the definition of cDV singularity

M. Reid defines cDV singularity as follow in his paper "CANONICAL 3-FOLDS"
A point $p\in X$ of a 3-fold is called a compound Du Val point if for some section H throgh $P$, $P\in H$ is a Du ...

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### Finiteness of rational double point

Let $(R,m)$ be a three dimensional complete local ring over a field $k$ of arbitrary characteristic and let $f\in R$ and $R/f$ is a rational double point.
My question is as follows.
Are there a ...

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### About the contractability

Let $X\subset \mathbb{P}^3$ be the surface defined by the equation $xy-zw=0$, and consider the curve $E \subset X$ defined by the equation $x=z=0$.
Question. Can $E$ be contracted to a point?

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### Canonical model and the existence of general hyperplane

A variety is a separated reduced scheme of finite type over an algebraically closed field $k$ not necessarily affine.
Let $X$ be a normal variety and $X$ has only one isolated singularity at $x\in X$ ...

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### A property of canonical singularity

Let $X$ be a normal variety with only one singularity at $x$ and $(X,x)$ is a canonical singularity i.e. $(X.x)$ satisfies $(i)$ and $(ii)$.
$(i)$ $(X,x)$ is a $\mathbb{Q}$ Goreinstein singularity.
$(...

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### Reference request The support of $f$-nef divisor

I'm seaching for a proof of the theorem below.
Do you know any reference?
Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...

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### Two different resolution of a three fold

Let $X\subset \mathbb{A}^4_\mathbb{C}$ be a three fold defined by the equation $xy-zw=0$
This variety has a singularity at origin of $\mathbb{A}^4_\mathbb{C}$
If we blow up this three fold in two ways ...

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### On the definition of the relative canonical divisor

Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...

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### Nice, concrete example of pl-flipping contraction

In a course I'm giving on the MMP, I am discussing the importance of Shokurov's notion of a pl-flipping contraction for showing that flips exist for arbitrary flipping contractions. Does someone have ...

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### Application of MMP in other branches of algebraic geometry

I'm learning minimal model program (MMP) recently. For a projective variety $X$, following MMP, we can do a sequence of birational transformations making $K_X$ nef or to a Mori fiber space.
My ...

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### Existence of a rational curve in the center of a birational contraction for symplectic singularities

Let $M$ be a holomorphically symplectic
complex manifold, and $f: M \to X$
a holomorphic, birational contraction to a Stein
variety $X$, contracting a subvariety $E$
to a point, and bijective outside ...

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### Finitely generated section ring of Mori dream spaces

Set-up: We work over $\mathbb{C}$. Let $X$ be a Mori dream space. Define, following Hu-Keel, the Cox ring of $X$ as the multisection ring
$$\text{Cox}(X)=\bigoplus_{(m_1\ldots,m_k)\in \mathbb{N}^k} \...

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### Minimal model program for toroidal pairs

Suppose $(X, \Delta)$ be a toroidal pair over $Z$ where $f:(X, \Delta) \rightarrow (Z, \Delta_Z)$ is a toroidal morphism (see https://arxiv.org/pdf/alg-geom/9707012.pdf sections 1.2, 1.3 for the ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 $(K_X + \Delta)$-...

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### 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 ...

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### 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 "...

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### 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 $\alpha$ ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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,...

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### 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$...

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### 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 $...

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### 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_{...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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}\...

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### 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 ...

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### 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 $p\in X$....

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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### One point compactification of the tangent bundle

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

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### 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 ...

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### 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,...

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### $K_X+B \equiv 0$ 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 ...

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### 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 ...