# Diagonal in projective space

This is exercise $2.15$ from Harris book "Algebraic Geometry: A First Course". Show that the image of the diagonal in $\mathbb{P}^{n} \times \mathbb{P}^{n}$ under the Segre map is isomorphic to the Veronese variety $v_{2}(\mathbb{P}^{n})$.

Would the idea is just map everything to itself and ignore the repeated monomials of degree $2$ for example if $n=1$ we get that the diagonal under the Segre mapping sends a point $([a : b],[a:b]) \mapsto [a^{2} : ab : ab : b^{2}]$. Now this almost looks like the $2$-Veronese map $\mathbb{P}^{1} \rightarrow \mathbb{P}^{2}$ given by $[s : t] \mapsto [s^{2} : st : t^{2}]$. So what I mean is simply ignore the repeated monomial $ab$ and map $[a^{2} : ab: ab : b^{2}]$ to $[a^{2} : ab : b^{2}]$. Would this work?

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## 1 Answer

You have exactly the right idea. I would formulate it slightly differently. If we continue with your example, we can write down a map $\mathbf P^2 \to \mathbf P^3$ as $$[x:y:z]\mapsto [x:y:y:z].$$ If we restrict this map to the Veronese embedded copy of $\mathbf P^1$, then we get an isomorphism onto the image of the diagonal under the Segre. This formalizes the idea that one can "ignore repeated monomials".

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