I have the following problems of morphisms between projective spaces, I would like to know if someone has any hint or knows how to solve them:

a) Suppose that there exists $ f: \mathbb{P}^r \rightarrow \mathbb{P}^s $ regular application, then $ r \leq s $ o $ f $ is constant

b) Let $ S: \mathbb{P}^n \times \mathbb{P}^n \rightarrow \mathbb{P}^{n^2 + 2n} $ the Segre application, consider $ v_2: \mathbb{P}^n \rightarrow \mathbb {P} ^ {\frac{n (n + 3)}{2}} $ the 2-Veronese's application. Prove that the diagonal $ \Delta \subset \mathbb{P}^n \times \mathbb{P}^n $ satisfies $ S (\Delta) \cong v_2 (\mathbb{P}^n) $


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    $\begingroup$ Hint: Are you aware of the equivalence of morphisms from a scheme $X$ to projective space with very ample line bundles on $X$? This makes the proof of (a) pretty short. For (b), this should be pretty straightforwards - where are you stuck? $\endgroup$ – KReiser Jun 4 at 1:45
  • $\begingroup$ I don't know the result of schemes that you mention, I do not know if you have any reference or another way to attack it? In 2 I'm stuck in some calculations, but I'm already trying them again Thank you @KReiser $\endgroup$ – Erick David Luna Núñez Jun 4 at 3:57
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    $\begingroup$ Hartshorne chapter II section 7 or EGAII section 4.4 are both sources for this material. Briefly, there is an equivalence between very ample line bundles $\mathcal{L}$ on $X$ and morphisms $f:X\to\Bbb P^n$ given in one direction by taking a basis $s_0,\cdots,s_n$ of $\mathcal{L}(X)$ and letting $f(x) = [s_0(x):\cdots:s_n(x)]$ and in the other direction by taking $\mathcal{L} = f^*\mathcal{O}(1)$. Now use the fact that any line bundle on $\Bbb P^n$ is $\mathcal{O}(m)$ and check which of these are very ample. $\endgroup$ – KReiser Jun 4 at 7:47

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