Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual lattice $M$ s.t. they give a non-negative inner product on the cone $\sigma$. Now, referring to Fulton Introduction to Toric Varieties page 5, he states that the semigroup $S_{\sigma}$ is generated by the dual vectors $e_{1}^{*}, e_{1}^{*} + e_{2}^{*}$, and $e_{1}^{*} + 2e_{2}^{*}$.
My question is why isn't it sufficient for $S_{\sigma}$ to be generated by the dual vectors $e_{1}^{*}$ and $e_{1}^{*}+2e_{2}^{*}$ which are normal to the generators for $\sigma$? Why do we need the extra dual vector $e_{1}^{*} +e_{2}^{*}$ which is needed in order to show that this toric variety is isomorphic to the variety $V(xz-y^{2})$?
Additionally, is there an easy/intuitive way to see that the spectrum Spec($\mathbb{C}[X,Y,Z]/[XZ-Y^{2}]$) gives a quadric cone?
Thanks!