  The geometry of the Fano scheme of maximal linear spaces contained in the
base locus of a pencil of quadrics has been studied by algebraic geometers when
the base field is algebraically closed. In this paper, we work over an
arbitrary base field of characteristic not equal to 2 and show how these Fano
schemes are related to the Jacobians of hyperelliptic curves. In particular, if
$B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$
is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component
of a disconnected commutative algebraic group $G = \picz(C) \dcup F \dcup
\pico(C) \dcup F'$, where $C$ is the hyperelliptic curve defined by the
discriminant form of the pencil. In the second half of this paper, we study
regular pencils of quadrics, where the hyperelliptic curve defined by the
discriminant is singular.
