Krdytkyu

up vote 12 down vote favorite 4 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 $$Big{z_{ij}z_{kl} - z_{il}z_{kj} hspace{1mm} Big| hspace{1mm} i,k = 0,ldots, n; hspace{2mm} j,l = 0,ldots,mBig}.$$ 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...