Projective pencil resolved by a point blowup

The holomorphic map $[Z_0:Z_1:Z_2]\mapsto [Z_1:Z_2]$ on ${\mathbb{CP}}^2\setminus \{[1:0:0]\}$ records the line through the missing point and the input point. Its graph closure is the incidence surface $Z_{1}b=Z_{2}a$ in ${\mathbb{CP}}^2\times {\mathbb{CP}}^1$, the blowup of a complex manifold at a point. Projection to $[a:b]$ extends the pencil holomorphically. On the exceptional divisor it is the identity map on the projective tangent directions.