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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.

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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
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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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Maybe I'm being dumb, but I don't quite follow. You say "Once you know that you have a section..." and conclude that you can find a section avoiding the singular locus of the fibers. But a section can never intersect the singular locus of a fiber, because at such a point the differential of f fails to be surjective. Am I missing something? (On the other hand, because of rational connectedness, Graber--Harris--Starr implies that there is always section.) –  Artie Prendergast-Smith Sep 29 '11 at 17:07
    
very cool. is all this explained in Kollar's book? for instance is it easy to see what is the "large enough" dimension? –  Olob Sep 29 '11 at 17:08
    
Yes, sorry Artie - I typed too quickly - answer amended. –  mdeland Sep 29 '11 at 17:29
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Actually it is Fano if n>2, since if n=2 it is an elliptic curve. Anyway, thanks a lot, very interesting! –  Olob Sep 29 '11 at 17:39
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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
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