TITLE: Sections of a fibration in intersections of quadrics
QUESTION [3 upvotes]: 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.

REPLY [1 votes]: As long as the fibers have large enough dimension, they are rationally connected.  Then something stronger than what you want follows from the comb smoothing technique developed by Mori, Koll\'ar and others (and explained in Koll\'ar's book Rational Curves on Algebraic Varieties) and the Graber-Harris-Starr (GHS).  Once you know that you have a section (GHS), that X is smooth, and that the fibers are rationally connected, then you can find a section through any specified finite number of points in (different, obviously) smooth fibers.  I first typed too quickly (thanks Artie) - the image of the section is automatically contained in the smooth locus of the map.  The complete intersection of two quadrics in P^4 is Fano, so rationally connected (anything of higher dimensional will also be)
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