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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 ...
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2 votes
1 answer
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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 ...
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13 votes
1 answer
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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 ...
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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 $...
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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?
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1 answer
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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. ...
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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 ...
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