I N T R O D U C T I O N

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.