= 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 $Z_1b=Z_2a$ 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.
Back to article page
