Questions tagged [minimal-model-program]

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

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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 ...
Csar Lozano Huerta's user avatar
22 votes
1 answer
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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 ...
Ariyan Javanpeykar's user avatar
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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 ...
Javier Álvarez's user avatar
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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 ...
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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 ...
unknown's user avatar
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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 ...
Gianni Bello's user avatar
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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 ...
John L.'s user avatar
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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 ...
Ben Webster's user avatar
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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 ...
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9 votes
1 answer
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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 ...
Ste3an's user avatar
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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 ?...
Pedro Montero's user avatar
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References for the minimal model program

What are some references for a beginning graduate student in algebraic geometry to learn about the minimal model program? I'm not thinking about entering this field, but rather I just want to know ...
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rational effective implies effective?

Let $X$ be a weak del pezzo surface, I Wonder whether the following statment is true: Let $L$ be a line bundle on $X$, then $h^0(L)=0$ implies $h^0(nL)=0$ for all $n\geq 1$.
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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 ...
Clay Cordova's user avatar
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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 ...
Kim's user avatar
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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 ...
Hydrogen's user avatar
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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 ...
K Kim's user avatar
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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 $...
Dimitri Koshelev's user avatar
7 votes
0 answers
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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 "...
Kim's user avatar
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Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program

My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology. Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...
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6 votes
1 answer
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Advantage of discrepancy

In the definition of Minimal model of projective variety, some authors use of discrepancy, and some others omit this condition. I am wondering to know the advantage of discrepancy In the definition of ...
Julien K's user avatar
6 votes
1 answer
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Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth. We know that ...
Omprokash Das's user avatar
6 votes
1 answer
453 views

Picard number of a general fiber of a fiber contraction

Suppose in the last step of a MMP, we obtain a Mori fiber space $f: X \to Z$, and let $F$ be a general fiber of $f$, then is the Picard number $\rho(F)$ of $F$ equal to $1$? Notice that the relative ...
Li Yutong's user avatar
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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 ...
Joaquín Moraga's user avatar
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Bogomolov–Miyaoka–Yau inequality for minimal varieties with intermediate Kodaira dimension $0<\kappa (X)<\dim X$

From the differential geometric proof of Yau and the algebraic proof of Miyaoka for minimal varieties of general type $\kappa (X)=\dim X$, we know that $$(-1)^nc_1^n(X)\leq (-1)^n\frac{2(n+1)}{n} c_1^...
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6 votes
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902 views

Restriction of the Canonical Divisor $K_X$ to a general fiber

Let $\ f:X\to Z$ be a surjective morphism between two smooth projective varieties with connected fibers $(f_*\mathcal{O}_X=\mathcal{O}_X)$. Let $F$ be a general fiber of $f$ and $\mbox{dim } F<(\...
Omprokash Das's user avatar
6 votes
0 answers
576 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 ...
Rogelio Yoyontzin's user avatar
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653 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 ...
Ben Webster's user avatar
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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 ...
Stefano's user avatar
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2 answers
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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 ...
Eugene Eisenstein's user avatar
5 votes
1 answer
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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 ...
Jesus Martinez Garcia's user avatar
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 ...
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5 votes
1 answer
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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). We ...
Dubious's user avatar
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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 $\...
Dima Sustretov's user avatar
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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 ...
Jérémy Blanc's user avatar
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0 answers
230 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, ...
numberjedi's user avatar
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0 answers
121 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 (...
Joaquín Moraga's user avatar
4 votes
1 answer
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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 $...
Pierre MATSUMI's user avatar
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 $X$.
fds's user avatar
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1 answer
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Bertini's type theorems over imperfect fields

Let $X$ be a projective variety over an imperfect (hence infinite and char(k)=p>0) field $k$. If the local rings of $X$ are all regular, then can we say that a general hyperplane section $H$ is also ...
Omprokash Das's user avatar
4 votes
1 answer
697 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 ...
Li Yutong's user avatar
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4 votes
1 answer
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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} \...
OrdinaryAttention's user avatar
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 ...
Jérémy Blanc's user avatar
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 ...
Misha Verbitsky's user avatar
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 ...
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4 votes
0 answers
142 views

Deminormal and Gorenstein

Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein? for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf
Kumar's user avatar
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A question about the dimension of a relatively ample divisor

Suppose $f: Y \to Z$ is a projective morphism of smooth varieties with connected fibers. If an effective divisor $H$ on $Y$ is relatively ample over $Z$, and $\dim Y >\dim Z$, is $h^0(Y, mH)>1$ ...
Li Yutong's user avatar
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4 votes
0 answers
170 views

Kuranishi family and smoothing of Calabi-Yau n-fold

Consider $X$ be a Calabi-Yau n-fold with at most one ordinary double point singularity. Suppose $X$ is smoothable. Then it is known that the Kuranishi family of $X$ is a smoothing of $X$. Now, ...
Larue's user avatar
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0 answers
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Example of a non-algebraic singularity II

In an answer of this MO question, Frank Loray constructed an example of analytic singularity which is not algebraic. On the other hand, as I learned from one of Joël's comments in that question, ...
HYL's user avatar
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4 votes
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A theorem about log Calabi-Yau pairs

Let $X$ be a normal variety with $\mathbb Q$-Cartier divisor $D$, such that $K_X+D$ is $\mathbb Q$-Cartier. Let $(X,D)$ is log Calabi-Yau pair, i.e, $K_X+D\sim_\mathbb Q0$. (For example take $ X$ be a ...
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