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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