Questions tagged [minimal-model-program]
minimal model program is part of the birational classification of algebraic varieties.
133 questions
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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, ...
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Quasi-projectivity of the moduli space of Kahler-Einstein Fano varities and vanishing Lelong number
Chi Li, Xiaowei Wang, Chenyang Xu proved the Quasi-projectivity of the moduli space of smooth Kahler-Einstein Fano manifolds. My question is about when central fibre $X_0$ along Kahler-Einstein Fano ...
user21574
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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 (...
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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 ...
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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 ...
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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 $\mbox{dim } F<(\...
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$L^2$ extension theorem
Is there an Ohsawa-Takagushi $L^2$-Extension theorem for Kahler manifolds? For projective varieties Siu-Paun proved:
Let $\pi \colon X \to \mathbb D$ be a projective family of $n$-folds and $X_0$ be ...
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Top self-intersection of the canonical divisor of a terminalization
Let $(X,\Delta)$ be a $n$-dimensional log canonica model,
and let $Y\rightarrow (X,\Delta)$ be a log terminalization
(meaning that $Y$ is the model obtained when we take a log resolution $\pi \colon ...
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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 ...
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Finiteness of models around a non-pseudo-effective ray
Let $(X,\Delta) $ a klt pair and $\rho $ a numerical class of divisor that is not contained in the pseudo-effective cone. Let $(X,\Delta_i)$ be a sequence of klt pairs such that $K_X+\Delta_i$ is not ...
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holomorphic fiber space when fibers are of general type
Let $\pi:X\to S$ be a holomorphic fibre space, then from birational geometry, the canonical divisor of general fibers are ample or trivial. When fibers are of general type?
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How can I describe in explicit geometric terms the (in general non-complete) linear system?
Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...
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Decompose a big divisor as nef big divisor and effective divisor
Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a $\mathbb{Q}$-...
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A question about potentially birational divisor
I am reading the paper "On the birational automorphisms of varieties of general type", and I have a question about the property of potentially birational divisor.
Definition (potentially birational ...
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Some examples where the plurigenera are nonconstant, when the fibres have worse singularities than canonical
Let start with a definition
Invariance of plurigenera: Choose $m$ large enough so that $mK_F$ has a non-zero global section for some fibre $F$. For any fibre $F$, we have $K_F = K_{X/D}~_{|F}$. So ...
user21574
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Run MMP between varieties of isomorphic in codimension 1
Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with ...
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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 ...
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Some questions on Kontsevich's moduli space
Motivation: Work of Eisenbud, Harris, and Mumford shows that
$\mathcal M_g$ is of general type when $g≥24$. Moreover, by Logan's function $f(g)$ , $\overline {\mathcal M_{g,n}}$ is of general type for ...
user21574
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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 ...
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Properties of extreme rays
Let $X$ be a projective variety over $\mathbb{C}$, then the effective curves module the numerical equivalence form a cone. For my understanding, if the extreme ray $[C]$ has the property that $[C]\...
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$C^\infty$-curvature of Kawamata's singular hermitian metric
Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka fibration. Consider the following singular hermitian metric $$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}$$ ...
user21574
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Why are the different definitions of minimal model equivalent?
I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:
$K_X$ is nef.
Every birational morphism from $X$ ...
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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 ...
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Finite generation of certain $\mathcal{O}_X$-algebra
It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the $\mathcal{O}_X$-...
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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 ?...
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Trivial canonical bundle
Let $X$ be a complex compact Kähler minimal surface with zero algebraic dimension and $H^2(X,\mathcal{O}_X) \ne 0$. We know that according to Enriques–Kodaira classification, $X$ is either a torus or ...
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Kawamata-Log-Terminal pairs
Let $p_1,...,p_n\in\mathbb{P}^3$ be general points, and let $\Delta\subset\mathbb{P}^3$ be a general surface of degree $d$ with points of multiplicity $m_i$ at $p_i$ for $i = 1,...,n$.
Consider the ...
user47036
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Controlling singularities on log mmp
Suppose all my varieties are complex threefolds $X\rightarrow Y$ over some smooth base curve germ $Y$. We can assume the fibres are Del Pezzo surfaces with generic smooth fibre.
If I do (relative) ...
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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 ...
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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 ...
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Minimal semistable model for K3-surfaces.
I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...
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On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
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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 ...
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