Questions tagged [minimal-model-program]
minimal model program is part of the birational classification of algebraic varieties.
137 questions
2 votes
0 answers
63 views
Existence of effective anti-ample divisors for birational morphisms
Let $(X, \Delta)$ be a (possibly non $\mathbb{Q}$-factorial) klt pair. Then is it always possible to find a log resolution $\pi: X' \to X$ of $(X, \Delta)$ such that the exceptional locus of $\pi$ ...
- 363
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
0 votes
1 answer
162 views
On the existence of flipping curves over birational base
Let $\pi: X' \to X$ be a birational morphism, where $X$ is $\mathbb{Q}$-factorial. Suppose $R=\mathbb{R}_+[C]\subset \overline{NE}(X'/X)$ is an extremal ray (not necessarily $K_{X'}$-negative) over $X$...
- 363
3 votes
1 answer
274 views
Mori dream surfaces with only contractions to projective spaces are del Pezzo
We work over the complex numbers, and let $X$ be a Mori dream surface. Suppose that all contractions of $X$ given by the rays of the Nef cone of $X$ are to $\mathbb{P}^2$ or to $\mathbb{P}^1$. Claim. $...
- 31
3 votes
0 answers
126 views
Minimal model program and reduction mod $p$
Let $(X,\Delta)$ be a projective klt pair over $\mathbb{C}$, and let $(X_R,\Delta_R)$ be a model of $(X,\Delta)$ over a finitely generated $\mathbb{Z}$-algebra $R\subseteq \mathbb{C}$. For any point $...
- 135
0 votes
0 answers
162 views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 3,854
2 votes
1 answer
238 views
Sequence of MMP with scaling cannot be isomorphism
Let $(X,B)$ be a projective klt pair, H is an $\mathbb{R}$-divisor s.t. $K_X+B+H$ is nef. Suppose that we can run $(K_X+B)$-MMP with scaling $H$(that is, the flip exists) and denote $\alpha_i:X \...
- 21
1 vote
0 answers
90 views
Discrepancy of general element of linear system
Let $X$ be a normal scheme and $|D|$ a linear system on $X$. In "Singularity of Minimal Model Program" by Janos kollar p249, it says, If $X$ is a variety over $\mathbb{C}$, and $E_j$ ...
- 368
0 votes
0 answers
124 views
Contraction of extremal ray on a smooth projective threefold
I have some issues about understanding the contraction of extremal ray in a concrete situation: Let $\mathcal{E}=\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1\times \...
- 41
2 votes
1 answer
191 views
Blow up of terminal singularity and canonical singularity
A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if $(i)$ it it is $\mathbb{Q}$-Gorenstein. and $(ii)$For any resolution of singularity $F:Y\rightarrow X$, $K_Y-f^*K_X>...
- 368
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
1 vote
0 answers
145 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 ...
- 368
2 votes
0 answers
146 views
Finiteness of rational double point
Let $(R,\mathfrak{m })$ 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 ...
- 368
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
2 votes
0 answers
143 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 $x\in X$ ...
- 368
2 votes
1 answer
211 views
A property of canonical singularity
Let $X$ be a normal variety with only one singularity at $x$ and $(X,x)$ is a canonical singularity i.e. $(X.x)$ satisfies $(i)$ and $(ii)$. $(i)$ $(X,x)$ is a $\mathbb{Q}$ Goreinstein singularity. $(...
- 368
2 votes
0 answers
127 views
Reference request The support of $f$-nef divisor
I'm seaching for a proof of the theorem below. Do you know any reference? Consider $f:X\rightarrow Y$ projective birational map between normal varieties and $\mathbb{R}$ cartier divisor $D$ whose ...
- 368
0 votes
1 answer
252 views
Two different resolutions of a three fold
Let $X\subset \mathbb{A}^4_\mathbb{C}$ be a three fold defined by the equation $xy-zw=0$ This variety has a singularity at origin of $\mathbb{A}^4_\mathbb{C}$ If we blow up this three fold in two ways ...
- 368
2 votes
0 answers
364 views
On the definition of the relative canonical divisor
Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...
- 293
1 vote
0 answers
139 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 ...
- 2,148
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
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
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} \...
2 votes
0 answers
198 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 ...
- 363
1 vote
0 answers
128 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 ...
- 4,265
2 votes
2 answers
273 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 ...
- 9,090
1 vote
0 answers
103 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 ...
- 363
1 vote
1 answer
221 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 (...
- 295
1 vote
0 answers
149 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 ...
- 363
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
3 votes
1 answer
330 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)$-...
- 10.7k
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
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
2 votes
1 answer
226 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$ ...
- 51
2 votes
1 answer
271 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 ...
- 9,090
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
1 vote
0 answers
237 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)$$ is an isomorphism, where $i:S\...
- 9,090
1 vote
0 answers
91 views
On the b-nefness of the moduli part of canonical bundle formula
I have recently been wondering about the existence of a canonical bundle formula in the following situation and am not sure how to proceed. Suppose $(X,B) \xrightarrow{f} Y$ is a fibration where $ (X,...
- 363
1 vote
0 answers
140 views
Canonical covering stack of a flop
In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...
- 439
2 votes
1 answer
198 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 $...
user114666
2 votes
0 answers
248 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_{...
- 363
2 votes
0 answers
158 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 ...
- 363
5 votes
1 answer
552 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
1 vote
1 answer
478 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 ...
user105074
3 votes
0 answers
783 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
197 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
1 vote
1 answer
576 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 ...
- 1,399
1 vote
0 answers
111 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}\...
- 2,012
1 vote
0 answers
248 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 ...
- 363
1 vote
1 answer
258 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 $p\in X$....
user61586