# Sections of a fibration in intersections of quadrics

Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces of dimension $n$.

Suppose that generically the quadrics are smooth and the singular ones have at most corank 1.

It is easy to see that such an object always has a rational section Y->X. Does it always have a regular section? Maybe for $n$ big enough? By regular I mean a section with no intersection with the singular locus of the fibers.

-
Do you want the quadric hypersurfaces to assemble into a flat family, or do you just want (geometric? complex?) fibers to be isomorphic to such an intersection? –  S. Carnahan Sep 29 '11 at 15:12
Can you count the rational sections? If there is a family of them, the answer is presumably yes, no? –  Will Sawin Sep 29 '11 at 15:44
Yes, I want a flat family of quadrics over a ruled surface $S\to Y$, whose relative intersection gives $X$ –  Olob Sep 29 '11 at 17:10

Just to expand slightly on Artie's comment: if $C$ is a section, then $C.F=1$ for all fibers $F$. If $F$ is singular at $P$ with multiplicity $m$ and $C$ passes through $P$, then the local intersection number $(C.F)_P$ is at least $m$. Then $1\ge m$, contradiction. –  inkspot Sep 29 '11 at 19:48