1. THE UNIVERSAL PSEUDO-QUOTIENT FOR
A FAMILY OF SUBVARIETIES
In this section we summarize the content of §§2-6, the main body
of this paper, in which we prove Main Theorem 0.2, Theorem 0.3 and
Corollary 0.11. Indeed, Main Theorem 0.2 and Corollary 0.11 are
immediate consequences of Theorem 0.3. Thus we spend the whole
§§2-6 for the proof of Theorem 0.3. (As to precisely how much are
needed from materials in §§2-6 for the proof of Main Theorem 0.2,
Corollary 0.11, see 1.5.)
The aim of this section is twofold; to give a brief preview of the
proof of Theorem 0.3 (in 1.1 and 1.4), and to introduce the key notion
of universal pseudo-quotients (in 1.2), with providing a result which
validates the notion (Theorem 1.3). We omit minor details and put
our emphasis on clarifying the mutual logical dependence of the results
in §§2-6.
Based on the discussion 1.1, in 1.5 we make clear how the assump-
tion (0.1.2) are used in our proof.
1.0. Let X C P ^
¥N
be a non-singular non-degenerate projective
variety, dimX = n 4, satisfying the assumptions (0.1.1-2); namely,
for a general point z G X there exists an m-dimensional closed sub-
variety Sz C X which is a non-singular quadric subspace in P (see
(0.10.4), 0.15 for the terminologies), and
(1.0.1) (= (0.1.2))
3 3
m —n (n ^ 5,6,10), m —n + 1 (n = 5,6,10).
5 5
Our first task is to determine the structure of the normal bundles
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