(i) Let Q, Q' be two quadric surfaces in P3. Assume
Q to be smooth. Show that the general quadric of the pencil uQ+vQ',
(u: v) in P1 is smooth, and that there are t singular quadrics
in the pencil, 1 <= t <= 4.
Hint 1
(ii) Give an example of a pencil of quadric surfaces in P3 such
that every member of the pencil is a singular quadric surface.
Answer.
