All Questions
Tagged with projective-morphisms ag.algebraic-geometry
19 questions
28
votes
2
answers
3k
views
Must the composition of projective morphisms be projective?
The notion of a projective morphism in algebraic geometry is surprisingly subtle. It is not quite clear what the definition is! For example, the definition in EGA differs from that in Hartshorne. ...
- 3,857
21
votes
1
answer
970
views
Can you give an example of two projective morphisms of schemes whose composition is not projective?
Grothendieck and Dieudonné prove in $EGA_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and if $Z$ is separated and quasi-compact, or if ...
- 47.5k
13
votes
1
answer
863
views
Generalization of the rigidity lemma in birational geometry
Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...
user75933
7
votes
1
answer
943
views
Examples of non-projective morphisms with projective fibres
Let $X\to S$ be a morphism of noetherian schemes such that, for all $s$ in $S$, the morphism $X_s\to $ Spec $k(s)$ is projective.
Then it doesn't follow that $X\to S$ is projective in general. In ...
- 71
6
votes
0
answers
1k
views
Generalized Euler sequence on a projective scheme
Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{...
- 63.1k
5
votes
1
answer
444
views
Self-intersection of a Cartier divisor
Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f\colon X\rightarrow C$, where $C$ is a curve.
May we conclude that $D^{2}=0$?
user58604
4
votes
3
answers
450
views
Existence of a morphism between two toric varieties
Does there exist a morphism between the blow-up of $\mathbb{P}^3$ in four general points and $\mathbb{P}^1\times\mathbb{P}^1$? If not why?
user68440
3
votes
1
answer
297
views
Is an equivariant projective morphism equivariantly-projective?
Let everything be over $\mathbb{C}$. Consider two varieties $X,$ $Y,$ where $X$ is normal and $Y$ is affine,
having regular $\mathbb{C}^*$-actions and
a $\mathbb{C}^*$-equivariant projective morphism
$...
- 1,677
2
votes
3
answers
2k
views
Why there are two point at infinity on certain elliptic curve [closed]
In article Adams, W. W., & Razar, M. J. (1980). Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc, 41, 481-498. is said on ...
- 424
2
votes
1
answer
186
views
Decomposition of a morphism with positive dimensional fibers
It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a ...
- 3,779
2
votes
1
answer
173
views
Non-reducedness of schemes and projective morphisms(revisited)
This is a continuation of a question asked by me previously with some added hypothesis. Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible ...
- 503
2
votes
0
answers
73
views
Projectivization of cokernel of 2-term-Koszul-like morphism depends only on certain simple data?
Let $X$ be a scheme and $D \subset X$ an effective Cartier divisor on $X$. For any line bundle ${\mathcal L} \in \mathrm{Pic}(X)$ and any global section $s \in \Gamma(X,{\mathcal L})$, define
$$ Y({\...
- 81
2
votes
0
answers
309
views
Extension of a rational section of a projective bundle
Let us assume that we work over the complex field and let $X$ be a smooth projective variety and $\pi: P \to X$ a projective bundle (i.e. a fibration in projective spaces of constant dimension). Let $...
- 3,779
1
vote
1
answer
99
views
Nonreducedness of schemes and projective morphisms
Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible subscheme. Suppose that the natural projection map $pr_2:W \to Y$ is surjective on the ...
- 503
1
vote
1
answer
219
views
Extending locally free sheaves and compatibility with fibers
Let $X$ be a smooth, projective variety over an algebraically closed field $k$ (of characteristic zero), $B$ a connected, noetherian scheme (possibly non-reduced) and $U$ an open subscheme of $X \...
- 2,323
1
vote
2
answers
404
views
Fibrations on blow-ups of $\mathbb{P}^2$
Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$.
Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...
user68440
1
vote
0
answers
168
views
Cohomology of a stratified projective bundle
Let $S$ be a smooth algebraic variety, and suppose $X\to S$ is a smooth morphism of schemes such that the geometric fibers are all projective spaces. Let us suppose that the dimension of the fibers is ...
- 3,779
1
vote
0
answers
157
views
The morphisms induced by two Cartier divisors
Let X be a projective variety. We consider two Cartier divisors $D,E$ such that $E\geq D$ and the relative morphisms
$\phi_D: X - - -> \mathbb{P}(H^0(X, O_X(D))^*)$ and $\phi_E: X- - -> \mathbb{...
0
votes
1
answer
323
views
Morphisms contracting a family of curves
Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
...
user58018