We work over an algebraically closed field k which, unless otherwise
noted, is assumed of characteristic zero; char k = 0. k is also assumed
to have an uncountable cardinality.
Let P := P ^ denote the TV-dimensional projective space over the
base field fc, which we will fix throughout, and X a non-singular n-
dimensional closed subvariety of P; i: X P, n 4.
The purpose of this paper is to determine the structure of such
X which is swept out by its quadric subspaces, i.e., its subvarieties
that are of degree 2 in P. We prove that if the dimension m of those
quadrics per n dimX is suitably large (see (0.1.2)), then X admits
either a linear or a quadric fiberspace structure (Theorem 0.3). In
particular, we prove that if those quadrics in X furthermore share a
common point to pass, then X is either a linear or a quadric subspace
(Main Theorem 0.2).
0.1. For a closed subscheme Z of P, the scheme theoretic linear
span (or simply the linear span) Z of Z in P is the smallest linear
subspace of P which contains Z as a closed subscheme.
We assume the following conditions (0.1.1-2):
(0.1.1) For a general closed point z G X, there exists an Tri-
dimensional non-singular closed subvariety Sz C X which passes
through z and is a quadric subspace in P; Sz Q™s^ i.e., a subvari-
ety Sz C X such that SZ is of dimension m + 1; Sz = P
m + 1
and Sz is a non-singular degree 2 hypersurface of Sz (it does not
Received by the editor August 10, 2000, and in revised form January 22, 2001.
The first author was supported by NSF Grant DMS-9800807.
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