Questions tagged [minimal-model-program]
minimal model program is part of the birational classification of algebraic varieties.
137 questions
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 ...
- 2,182
22 votes
1 answer
960 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 ...
- 9,697
19 votes
2 answers
6k views
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 ...
- 1,345
15 votes
2 answers
2k views
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 ...
- 10.7k
13 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 ...
- 163
12 votes
3 answers
3k views
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 ...
- 1,160
12 votes
1 answer
994 views
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 ...
- 183
10 votes
0 answers
642 views
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 ?...
- 1,017
9 votes
2 answers
2k views
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 ...
- 45.5k
9 votes
2 answers
2k views
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 ...
user5117
9 votes
1 answer
540 views
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 ...
- 133
8 votes
2 answers
1k views
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 ...
- 1,557
8 votes
1 answer
371 views
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 $n\geq 1$.
- 2,012
7 votes
3 answers
1k 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 ...
- 2,097
7 votes
1 answer
645 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 ...
- 4,232
7 votes
1 answer
615 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 ...
- 361
7 votes
1 answer
678 views
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 ...
- 73
7 votes
1 answer
514 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 $...
- 1,843
7 votes
1 answer
624 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 ...
- 3,584
7 votes
0 answers
525 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 "...
- 4,232
7 votes
0 answers
609 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 ...
- 1,593
7 votes
0 answers
431 views
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 ...
user21574
7 votes
0 answers
1k 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<(\...
- 333
6 votes
1 answer
1k views
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 ...
- 333
6 votes
1 answer
652 views
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 ...
- 63
6 votes
1 answer
2k views
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 ...
- 333
6 votes
0 answers
610 views
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^...
user21574
6 votes
0 answers
639 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 ...
6 votes
0 answers
692 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 ...
- 45.5k
5 votes
1 answer
374 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 ...
- 625
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 ...
5 votes
1 answer
451 views
Contractibility of a curve on a surface
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?
- 368
5 votes
1 answer
1k views
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 ...
- 2,205
5 votes
1 answer
657 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 ...
user149914
5 votes
1 answer
541 views
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 ...
- 1,247
5 votes
1 answer
553 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 $\...
- 4,150
5 votes
0 answers
213 views
Flops connect minimal models of algebraic spaces?
According to a Kawamata's result, two projective minimal models of the same variety are connected through a sequence of flops. In particular, a birational map $f\colon X\to X'$ between Calabi-Yau ...
- 178
5 votes
0 answers
198 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 ...
- 7,916
5 votes
0 answers
285 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, ...
- 61
5 votes
0 answers
154 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 (...
- 1,593
4 votes
1 answer
3k views
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 $...
- 791
4 votes
1 answer
2k 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$.
- 427
4 votes
1 answer
592 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 ...
- 333
4 votes
1 answer
913 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 ...
- 3,584
4 votes
1 answer
208 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} \...
4 votes
1 answer
171 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 ...
- 7,916
4 votes
0 answers
160 views
Does relative base locus coincide with $\operatorname{Supp}\operatorname{coker} (f^*f_*\mathcal{O}_X(D)\rightarrow\mathcal{O}_X(D))$?
Let $f\colon X\rightarrow S$ be a proper morphism between normal varieties, and let $D$ be a Cartier divisor on $X$. We say that two Cartier divisors $D$ and $D'$ are linearly equivalent over $S$, ...
- 275
4 votes
0 answers
140 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 ...
- 9,419
4 votes
0 answers
164 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 ...
user168611
4 votes
0 answers
205 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
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