All Questions
18 questions
4
votes
0
answers
1k
views
Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
- 4,164
2
votes
0
answers
164
views
A question on Okounkov bodies
Let $X$ be an irreducible $n$-dimensional projective variety, and
$$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$
a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
user125056
2
votes
1
answer
209
views
Curves contracted by a rational map
Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...
user114666
2
votes
1
answer
181
views
Anti-canonical divisorial contractions of weak Fano $3$-folds
Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...
user114198
5
votes
0
answers
232
views
In search for examples concerning pushforward of nef divisors and lc-trivial fibrations
My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...
- 625
3
votes
2
answers
924
views
Rational maps and Kodaira dimension
Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$.
Assume that $Y$ is of general type. May we conclude then that $X$ ...
- 8,998
3
votes
2
answers
968
views
Rationality of conic bundles
Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...
- 8,998
2
votes
0
answers
341
views
Lefschetz type theorems for big and nef divisors
Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that the restriction map $Pic(X)\rightarrow Pic(D)$ is an isomorphism over $\mathbb{Q}$.
Under which ...
user82886
1
vote
1
answer
240
views
Divisor class group of quartic surfaces
Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.
What can we say about the singularities of $X$?
user97096
1
vote
0
answers
70
views
Moving curves and small transformations
Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...
- 141
1
vote
0
answers
312
views
Cone of moving curves
Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...
- 141
1
vote
1
answer
273
views
Big divisors in family
Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ ...
- 625
1
vote
1
answer
1k
views
Pull-back of the canonical divisor via a rational map
Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many ...
user97096
1
vote
1
answer
242
views
Number of conditions imposed by fat points to a linear system
Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...
user97096
2
votes
1
answer
511
views
Rigid effective divisors
Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\...
user77919
2
votes
3
answers
2k
views
Movable Divisors
Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?
user56259
0
votes
1
answer
248
views
Big divisors and small transformations
Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:
$-K_Y$ is ...
user49214
3
votes
2
answers
5k
views
Big and Nef divisors
In Example 2.2.19 of
Lazarsfeld, Positivity in Algebraic Geometry I,
I found the following statement:
Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...
user47036