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6
votes
1answer
190 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 $D^{2}=0$?
14
votes
1answer
330 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 ...
7
votes
3answers
293 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?
3
votes
2answers
225 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 ...
1
vote
1answer
163 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. ...
0
votes
3answers
486 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 ...
2
votes
1answer
153 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 ...
1
vote
1answer
85 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 ...
2
votes
0answers
128 views

Jacobian Conjecture, Cubic-Keller maps

I have recently read an interesting article about the Jacobian Conjecture, in particular the reduction to the case $f(x) = x + A(x)^3$. I was wondering about codimension one divisors on $Y = A^n$. ...
2
votes
0answers
163 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 ...
3
votes
0answers
518 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{...
21
votes
2answers
2k 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. ...