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# 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 ...
• 241
1 vote
0 answers
35 views

### 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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2 votes
0 answers
71 views

### 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 ...
• 241
3 votes
1 answer
228 views

### 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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2 votes
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84 views

### 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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2 votes
1 answer
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1 vote
0 answers
108 views

### 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 ...
• 2,108
7 votes
1 answer
496 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 ...
• 313
4 votes
0 answers
87 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 ...
• 9,145
4 votes
1 answer
144 views

2 votes
0 answers
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• 4,050
1 vote
1 answer
390 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 ...
3 votes
0 answers
517 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 ...
• 7,672
1 vote
1 answer
396 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 ...
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1 vote
0 answers
85 views

2 votes
1 answer
367 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
516 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
689 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 ...
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2 votes
0 answers
75 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 ...
• 335
2 votes
2 answers
434 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
860 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 ...
• 9,437
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,...
• 1,801
5 votes
1 answer
301 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 ...
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1 vote
1 answer
222 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 ...
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