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I have three questions of increasing generality that relate to an exercise in Chapter 1, Section 3 of Hartshorne's Algebraic Geometry concerning the $d$-uple embedding. I'd be happy to have any of them answered. $k$ is an algebraically closed field.

  1. For a fixed $n, d$ let $M_0, ..., M_n$ be the monomials in $n+1$ variables of degree $d$. The $d$-uple embedding $\varphi : \mathbb{P}^n \to \mathbb{P}^N$ is defined via $P \mapsto (M_0(P), ..., M_N(P))$. If $\theta$ is the $k$-algebra homomorphism $k[y_0,...,y_N] \to k[x_0,...,x_n]$ via $y_i \mapsto M_i$, then clearly $\text{im } \varphi$ is contained and Zariski dense in $Z(\ker \theta)$. How to show that in fact $\text{im } \varphi = Z(\ker \theta)$, i.e. $\text{im } \varphi$ is a variety?

  2. Whenever we have a map between affine (or projective) spaces given by componentwise polynomial (or monomial, of the same degree) maps, the image of the morphism will be contained and Zariski dense in the kernel of the corresponding $k$-algebra homomorphism. However, it need not be true that the image of the morphism is itself a variety in these circumstances (e.g. $(a,b) \mapsto (a, ab)$). Are there sufficient conditions to guarantee that the image corresponds precisely with the kernel of the $k$-algebra homomorphism that apply to the situation in (1)?

  3. Are there any sufficient conditions of a general morphism of varieties guaranteeing the image will be a variety that apply to the situation in (1)?

Feryll
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    Projective varieties are proper, so the image of a projective variety under any morphism is always closed. – Nefertiti Sep 04 '17 at 08:39
  • Huh, that's a very useful general fact. Unfortunately, Hartshorne doesn't seem to go into that until well into the chapter on schemes... – Feryll Sep 04 '17 at 21:12
  • If you want to see a proof in the language of varieties, you can look at Volume 1 of Shafarevich, Chapter I, Section 5.2, Theorem 2 (in the second edition). – Nefertiti Sep 06 '17 at 09:52

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