In the book of Harris Algebraic Geometry, a First Course, exercise 2.29, it is asked to show that the composition of $1\times \nu: \mathbb{P}^1\times\mathbb{P}^1 \to \mathbb{P}^1\times\mathbb{P}^2$ where $\nu$ is the Veronese embedding with the Segre embedding $\sigma:\mathbb{P}^1\times\mathbb{P}^2 \to \mathbb{P}^5$ can be represented (after identification of $\mathbb{P}^1\times\mathbb{P}^1$ with its image the quadric $Q: Z_0Z_3-Z_1Z_2=0$ in $\mathbb{P}^3$ by the Segre mapping), by the map $$\phi: [Z_0,Z_1,Z_2,Z_3]\to [F_0(Z),\cdots,F_1(Z)]$$ where $(Q,F_0,\cdots,F_5)$ is a basis of the 7 dimensional space of homogenous quadratic polynomials that contains the line $L: Z_0=Z_1=0$ of $Q$.
This seems strange to me because $\phi$ is not even defined on the line $L$. What did I miss ?