# Morphism between projective spaces

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 $$S (\Delta) \cong v_2 (\mathbb{P}^n)$$

Thanks

• 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? – KReiser Jun 4 at 1:45
• 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 – Erick David Luna Núñez Jun 4 at 3:57
• 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. – KReiser Jun 4 at 7:47