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19 votes
2 answers
5k 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 ...
Javier Álvarez's user avatar
9 votes
2 answers
1k 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 ...
user avatar
9 votes
1 answer
502 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 ...
Ste3an's user avatar
  • 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 ...
DCT's user avatar
  • 1,537
7 votes
1 answer
567 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 ...
Kim's user avatar
  • 4,164
7 votes
1 answer
549 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 ...
Hydrogen's user avatar
  • 361
7 votes
1 answer
424 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 $...
Dimitri Koshelev's user avatar
7 votes
0 answers
571 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 ...
Joaquín Moraga's user avatar
6 votes
1 answer
617 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 ...
Julien K's user avatar
6 votes
1 answer
1k 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 ...
Omprokash Das's user avatar
6 votes
1 answer
519 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
  • 3,472
6 votes
0 answers
993 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
5 votes
1 answer
320 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 ...
Stefano's user avatar
  • 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 ...
Eugene Eisenstein's user avatar
5 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 ...
Omprokash Das's user avatar
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 ...
Jesus Martinez Garcia's user avatar
5 votes
1 answer
452 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 $\...
Dima Sustretov's user avatar
5 votes
0 answers
159 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 ...
fgh's user avatar
  • 178
5 votes
0 answers
171 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
4 votes
1 answer
527 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 ...
Omprokash Das's user avatar
4 votes
1 answer
785 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
  • 3,472
4 votes
1 answer
167 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} \...
OrdinaryAttention's user avatar
4 votes
1 answer
134 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
102 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
136 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 ...
user avatar
4 votes
0 answers
221 views

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
  • 3,472
4 votes
0 answers
628 views

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 ...
user avatar
4 votes
0 answers
172 views

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 ...
Li Yutong's user avatar
  • 3,472
3 votes
1 answer
255 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?
George's user avatar
  • 328
3 votes
1 answer
569 views

A covering lemma of Kawamata

In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is Let $X$ be a non-singular projective variety, and $D$ be a divisor with ...
Li Yutong's user avatar
  • 3,472
3 votes
1 answer
500 views

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 ...
Li Yutong's user avatar
  • 3,472
3 votes
1 answer
254 views

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 $\mathcal{O}_X$-...
Li Yutong's user avatar
  • 3,472
3 votes
1 answer
291 views

Movable divisor with base locus on a hyperkahler variety

I'm looking for an example of the following: $X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$); $D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of $D$ is a ...
Ennio Mori cone's user avatar
3 votes
0 answers
135 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,...
Dimitri Koshelev's user avatar
3 votes
0 answers
78 views

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 ...
Joaquín Moraga's user avatar
2 votes
2 answers
208 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 ...
Puzzled's user avatar
  • 8,998
2 votes
1 answer
218 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 ...
Puzzled's user avatar
  • 8,998
2 votes
1 answer
727 views

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 $X$ ...
user17314's user avatar
2 votes
1 answer
451 views

Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...
Omprokash Das's user avatar
2 votes
1 answer
222 views

Flatness of Fano Contractions

In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the ...
user avatar
2 votes
1 answer
679 views

A Decomposition for Iitaka fibration

Let $\pi: X\to Y$ be an Iitaka fibration of projective varieties $X,Y$, then is there always the following decomposition $$K_Y+\frac{1}{m!}\pi_*\mathcal O_X(m!K_{X/Y})=P+N$$ where $P$ is ...
user avatar
2 votes
1 answer
915 views

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]\...
Li Yutong's user avatar
  • 3,472
2 votes
1 answer
267 views

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)(...
Omprokash's user avatar
  • 165
2 votes
1 answer
171 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 $...
user avatar
2 votes
0 answers
244 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 $...
Don's user avatar
  • 293
2 votes
0 answers
214 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_{...
anonymous's user avatar
  • 335
2 votes
0 answers
119 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 ...
anonymous's user avatar
  • 335
2 votes
0 answers
79 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 ...
anonymous's user avatar
  • 335
2 votes
0 answers
674 views

Small contractions as blow ups

To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here. I am trying to learn a bit about birational morphisms:...
harajm's user avatar
  • 121
1 vote
2 answers
550 views

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 $\mathbb{Q}$-...
Li Yutong's user avatar
  • 3,472