2 - Dept. Math, Hokkaido Univ. EPrints Server

After finishing the present paper we found that h.II.Paranjape and
V.Srinivas gc~t the same result (= Proposition 8) in the following
paper:
Self maps of homo~eneous aspcas.
Invent.rnath.98. 123-4~14(1989).
Cur method of the proof is essencialy same as the one by them
except for the last part of the proof. The last part seems to be
most cGmplicated one. That is why we have the ar9ument in any
characteristic
-
(T 2). We must study the property of the homomorphism
*
TQ f TI on the branched locus R in details.
G:
Yotation. Basically vie use cust~lmary terrninolo2ies of
algebraic geometry. -1 variety means a separated, reduced irreducible
al2ebraic k-scheme where li is an algebraicall>- closed field of any
characteris~ic. 0 (1) denotes the line bundle correspond in^ to the
P "
hyperplaes in pn and in case of n=l it is abbri~i~xted simp1:- as
O ( 1 ) very often. Khen a vector bundle E on a variety is generated
b~- its glcbc21 sections, for the simplicity we s.3~ that E is GS.
For a vsriet:: X and a closed subscheme which is locally complete
intersection in S, ',
-'I- / S means the normal b~lndle of T in S.