All Questions
Tagged with minimal-model-program divisors
11 questions
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
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
1
vote
0
answers
202
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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\...
- 8,998
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
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 ...
- 7,722
1
vote
1
answer
215
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
2
votes
2
answers
515
views
Intuition behind Kawamata's definition of a relative movable Cartier divisor
I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]:
[1] Y. Kawamata, ...
- 3,022
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
4
votes
1
answer
2k
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 $...
- 781
4
votes
1
answer
1k
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