TITLE: Hartshorne Problem 1.2.14 on Segre Embedding
QUESTION [19 upvotes]: This is a problem in Hartshorne concerning showing that the image of $\Bbb{P}^n \times \Bbb{P}^m$ under the Segre embedding $\psi$ is actually irreducible. Now I have shown with some effort that $\psi(\Bbb{P}^n \times \Bbb{P}^m)$ is actually equal to $V(\mathfrak{a})$ where $\mathfrak{a}$ is the ideal generated  by the set of all monomials 
$$\Big\{z_{ij}z_{kl} - z_{il}z_{kj} \hspace{1mm} \Big| \hspace{1mm} i,k = 0,\ldots, n; \hspace{2mm} j,l = 0,\ldots,m\Big\}.$$


My main problem now is in showing that $\mathfrak{a}$ is actually equal to the kernel of the ring homomorphism $$\varphi : k[z_{ij}] \to k[x_0,\ldots,x_n,y_0,\ldots,y_m]$$
    that sends $z_{ij}$ to $x_iy_j$.


I have spent quite a few hours playing around with monomial orderings and trying to show that $\mathfrak{a} \supseteq \ker \varphi$ but to no avail. Of course the other inclusion is immediate. 
Is there anything I can do apart from playing around with monomial orderings to try and show that the kernel of $\varphi$ is equal to $\mathfrak{a}$? Perhaps maybe something along the lines of inducting on $n$, those this does not look promising.
Note: Please do not close this question; my question is different from the other questions on this site concerning the Segre embedding. Also, I can show my work concerning how I arrived at the conclusion that $V(\mathfrak{a}) = \psi(\Bbb{P}^n \times \Bbb{P}^n)$.

REPLY [5 votes]: It's easy to show the preimage under the Segre map of an algebraic set is an algebraic subset of $\mathbb{P}^n \times \mathbb{P}^m.$ If $V$ were reducible, we could write $V = V_1 \cup V_2,$ where $V_i\neq V$ are closed. Then $$ \mathbb{P}^n \times \mathbb{P}^m = S^{-1}(V) = S^{-1} (V_1) \bigcup S^{-1}(V_2)$$ where $S^{-1}(V_i)$ are closed. Picking $x_i \in V\setminus V_i$ we have $S^{-1}(x_i)\cap S^{-1}(V_i)=\emptyset$ so $S^{-1}(V_i) \subset S^{-1}(V)$ is a strict inclusion, contradicting that $\mathbb{P}^n \times \mathbb{P}^m$ is irreducible.