I would like to understand the flat deformation from the quadric surface $F_0=\mathbb{P}^1\times\mathbb{P}^1$ to $F_2$.
I think you deform from the smooth quadric $F_0$ to a singular quadric (A$1$ singularity) then resolve to get $F_2$, or something like that. Could someone help by an explicit computation please? Thank you.