All Questions
67 questions
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
1
vote
1
answer
174
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 $\alpha$ ...
- 31
1
vote
1
answer
820
views
Are terminal singularities $ \mathbb{Q}$-factorial?
The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need ...
- 335
1
vote
0
answers
90
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=\...
- 6,016
1
vote
1
answer
86
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 \...
- 11
1
vote
0
answers
113
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 ...
- 2,108
1
vote
0
answers
113
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 ...
- 3,730
1
vote
0
answers
65
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 ...
- 335
1
vote
0
answers
123
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 ...
- 335
1
vote
0
answers
202
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\...
- 8,998
1
vote
0
answers
69
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,...
- 335
1
vote
0
answers
216
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 ...
- 335
1
vote
0
answers
162
views
Compactifying morphisms and ample line bundles
Let $f:X\to Y$ be a projective morphism between two normal quasi-projective varieties, and $L$ a $f$-ample line bundle on $Y$. Then the claim is: There is a compactication $\bar{f}:\overline{X}\to\...
- 313
1
vote
0
answers
50
views
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(\...
- 1,833
0
votes
1
answer
325
views
On the birational equivalent class of algebraic surfaces with Picard number $1$
An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$.
Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...
- 828
0
votes
1
answer
700
views
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
0
votes
0
answers
89
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