= Rank-one sections of two hyperplane line bundles
{title2=$s=(\ell_1,\ell_2),\quad\operatorname{rank}\begin{pmatrix}a&b\\c&d\end{pmatrix}=1$}
A section of $\mathcal O(1)\oplus\mathcal O(1)$ on the <complex projective line> is a pair of homogeneous linear forms $\ell_1=aZ_0+bZ_1$, $\ell_2=cZ_0+dZ_1$. It vanishes at exactly one projective point precisely when its coefficient matrix has rank one, equivalently $ad-bc=0$ and the matrix is nonzero. Its kernel is then one line in $\mathbb C^2$. Equivalently, $s=v\ell$ for a constant nonzero vector $v$ and one nonzero linear form $\ell$. Rank two gives no zero, while rank zero gives the identically zero section.
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