Questions tagged [minimal-model-program]

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

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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 ...
7 votes
1 answer
477 views

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 ...
4 votes
0 answers
85 views

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 ...
4 votes
1 answer
137 views

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} \...
3 votes
0 answers
283 views

Nefness property for symplectic equivalency of Moishezon manifolds

Definition:Two symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are defined to be symplectically equivalent if there exists a diffeomorphism $\phi:X\to Y$ such that $\phi^∗ω_Y$ is in the same ...
2 votes
0 answers
91 views

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 ...
1 vote
0 answers
113 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 ...
2 votes
2 answers
187 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
59 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 ...
4 votes
1 answer
1k 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 $X$.
5 votes
2 answers
1k 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 ...
1 vote
1 answer
186 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 (...
7 votes
3 answers
931 views

Basepoints in the canonical system of algebraic surfaces

Let $X$ be a 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 ...
1 vote
0 answers
110 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
130 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
221 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 $(K_X + \Delta)$-...
3 votes
0 answers
281 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
529 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
393 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 "...
2 votes
1 answer
158 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 $...
1 vote
1 answer
158 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 $\alpha$ ...
12 votes
3 answers
4k 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 ...
2 votes
1 answer
191 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
127 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 ...
1 vote
0 answers
196 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
1 answer
623 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
206 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_{...
1 vote
0 answers
66 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
90 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
0 answers
105 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 ...
3 votes
0 answers
209 views

local description of $\mathbb{P}^2$-fibrations over $\mathbb{P}^1$

Let $X$ be a rational threefold (over the field of complex numbers) with terminal singularities. It is well-known that $X$ has only finitely many singular points $x_1,x_2, \ldots,x_n$. To be more ...
25 votes
5 answers
4k 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 ...
5 votes
1 answer
402 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
372 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 ...
2 votes
0 answers
478 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
157 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
370 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
84 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
213 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
192 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
302 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\...
22 votes
1 answer
854 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 ...
2 votes
1 answer
365 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
510 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 ...
2 votes
0 answers
74 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
431 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 $\mathbb P^1$?
2 votes
1 answer
387 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
0 answers
134 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
299 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
209 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 ...