Questions tagged [dominant-morphisms]
For questions related to dominant morphisms. A morphism $f:X→S$ of schemes is called dominant if the image of $f$ is a dense subset of $S$.
6 questions
0 votes
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97 views
Image of a dominant map contains an open set
Consider a dominant morphism of affine irreducible varieties, the image contains a non-empty open subset of the codomain. I know that this can be proved using Chevalley's theorem, but I was wondering ...
- 47
0 votes
1 answer
136 views
Function field and rational maps to the projective line correspondence
Let $V/k$ be a variety. There seems to be a well-known correspondence between non constant elements of the function field $K(V)$ and dominant rational maps $V\dashrightarrow\mathbb{P}^1$. Furthermore, ...
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1 vote
1 answer
160 views
Question about flat base-change
Let $\varphi \colon X\to S_2$ be a proper dominant morphism between varieties. (This varieties has the same dimension in my example if it helps) I have another morphism $\psi\colon S_1\to S_2$ which ...
- 672
2 votes
0 answers
318 views
Proof of Chevalley's theorem in Milne's book
I'm facing some difficulties into understanding the proof of the famous "Chevalley Theorem" in Milne's "Algebraic Geometry" (page 196) A regular morphism $\varphi:W\to V$ between ...
- 3,519
0 votes
1 answer
189 views
Dominant map is dominant in generic fiber
Suppose we have a dominant morphism of varieties $f : X \rightarrow Y$ over $Z$ (i.e we have two fibrations $X \rightarrow Z$ and $Y \rightarrow Z$ on which we can add all the properties needed as ...
- 1,081
2 votes
1 answer
264 views
Interior of the image of a morphism
Let $\phi:X\to Y$ be a morphism between irreducible quasiprojective varieties. If $\phi$ has a dense image in $Y$ can we conclude that its image has an interior? It really feels like it, but I couldn'...
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