All Questions
67 questions
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 ...
- 1,335
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 ...
user5117
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 ...
- 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 ...
- 1,537
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 ...
- 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 ...
- 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 $...
- 1,833
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 ...
- 1,595
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 ...
- 63
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 ...
- 313
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 ...
- 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<(\...
- 313
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 ...
- 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
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 ...
- 313
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 ...
- 2,215
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 $\...
- 4,070
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 ...
- 178
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 ...
- 313
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 ...
- 3,472
4
votes
1
answer
168
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
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 ...
- 7,722
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 ...
- 9,237
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 ...
user168611
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$ ...
- 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 ...
user21574
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 ...
- 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?
- 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 ...
- 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 ...
- 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$-...
- 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 ...
- 549
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,...
- 1,833
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 ...
- 1,595
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 ...
- 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 ...
- 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 $X$ ...
- 93
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 ...
- 313
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 ...
user21574
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]\...
- 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)(...
- 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 $...
user114666
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 $...
- 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_{...
- 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 ...
- 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 ...
- 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:...
- 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 $\mathbb{Q}$-...
- 3,472
1
vote
1
answer
193
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
1
answer
433
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