All Questions
Tagged with line-bundles projective-geometry
21 questions
2
votes
0
answers
242
views
Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
- 8,998
2
votes
1
answer
193
views
Linear system giving the projective embedding of the tangential variety
I was looking for a detailed explanation of a standard construction involving the projective tangential variety but I'm not able to find it anywhere, so maybe here some expert can enlight me on this ...
- 1,343
23
votes
5
answers
10k
views
Maps to projective space determined by a line bundle
The following should be pretty standard for any algebraic geometer.
Let $X$ be a compact complex variety, and let $L$ be a line bundle on $X$. We say $L$ is 'generated by global sections' if for ...
- 13k
2
votes
0
answers
677
views
Embeddings of Hirzebruch surfaces $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$
Let $X_n=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(n))$ be the $n-$th Hirzebruch surface. We know that for $d>0$ and higher $k>>0$ the linear system $$\mathcal{L}_{...
- 1,343
3
votes
2
answers
968
views
Rationality of conic bundles
Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...
- 8,998
2
votes
0
answers
164
views
A question on Okounkov bodies
Let $X$ be an irreducible $n$-dimensional projective variety, and
$$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$
a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
user125056
20
votes
1
answer
2k
views
Maps to projective space == line bundles; what do maps to weighted projective space correspond to?
A map from an algebraic variety $X$ to a projective space is the same thing as a globally generated line bundle on $X$. What geometric object on $X$ corresponds to a map to a weighted projective space?...
- 40.3k
3
votes
2
answers
924
views
Rational maps and Kodaira dimension
Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$.
Assume that $Y$ is of general type. May we conclude then that $X$ ...
- 8,998
1
vote
1
answer
1k
views
Pull-back of the canonical divisor via a rational map
Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many ...
user97096
2
votes
1
answer
209
views
Curves contracted by a rational map
Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...
user114666
2
votes
1
answer
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
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
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
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
7
votes
2
answers
608
views
Will (general points + small number of arbitrary points) impose independent condtions on plane curves?
It is well known that imposing vanishing at general points of $\mathbb P^2$ gives independent conditions on curves of degree $d$. Also, it is known that a small number ($\le d+1$) points always impose ...
- 1,509
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
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
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
2
votes
1
answer
1k
views
Line bundles, linear systems and normalization
One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on $\mathbb{P}^...
- 23
1
vote
0
answers
389
views
canonical model of a reducible curve
Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
- 3,779
0
votes
0
answers
251
views
Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...
- 1