= Projection from a point on a smooth quadric
{title2=$[x_0:x_1:x_2:x_3]\mapsto[x_1:x_2:x_0-x_3]$}
For $X=\{x_0x_1=x_2x_3\}$ and $P=[1:0:0:1]$, the displayed map is defined on $X\setminus\{P\}$. Its fibre over $[u:v:w]$ has equation $(u-v)\lambda+uw=0$ on an affine line. Away from $u=v$ the fibre is one reduced point. On that line it is empty except at $[0:0:1]$ and $[1:1:0]$, where it is a punctured ruling line, hence an affine line. The two exceptional lines are the <rulings of a smooth quadric surface> through $P$.
Back to article page
