Algebraic Curves and Projective Geometry by Edoardo Ballico, Ciro Ciliberto

By Edoardo Ballico, Ciro Ciliberto

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F. ~ ) = ( 1 1 , 1 2 , 1 3 , . . (~ ~) = ( 1 2 , 1 3 . . ) . Example. 8. 1. , I r) x~ = = set Xll Xl2... Xlr in particular, x ~ that for any ~ = y~ n>~o , y~ = Yll ... Yl r ; = I. The first assertion~ in 3,1(e) means tho monomials (x~ I A ~ ~ ) form a basis of R n. We apply the following general Lemma. ,ep be elements of an arbitrary T-group R. 2) if and only if the Gram determinant d e t ( < ei, ej >)i, j=1 , ... ,p equals 1. Rl ~ Proof. ,e p form a basis of a ~-subgroup R. ,ep} Therefore is invertible, Conversely, between the bases f & £ ) 4 ) , t , ~ a n d and A and A -I are both integral.

2, any irreducible element is an irreducible cons- tituent of then n ~n for some ~ R n ~ - - ~"r. Z In particular, is divisible by d e g ~ grading on R without loss of generality that d e g f Theorem. (b) For any and Yn in x2 n~O Rn x2(Y n) = O, and , and assume = 1. Thus, are summarized in the following. (a) The element ~ 2 ducible elements xn R Rn ~ 0 . 9). ,x n ad (x n) = o; Yn" In particular, for Yk (Xn) = xk (Yn) = 0 (d) For then n~ I k ~ 2. ~ of polyno- mials in indeterminates (Xn/ n ~ 1 ) ; similarly, R = l ~ 1 , y 2 .

26 as desired. /ZL/~ J, Let t be the decompositions of j'~ and I ~ ! " l J sums of irredu- into cible elements. ~! D. 7. 2. 2) equals S(~) I) + is the characteristic function of the sub- set ~ p ~ d ~ - -~ ~ . 5 (b) R(~),LL is a PSH-subalgebra of Clearly, any ~ ~ S(~') ~ ) R . 6, we see that the multiplication establishes an isomorphism of T-groups m : Since that @ R(nf ~R(~O). 5 (b)), it follows establishes an isomorphism of PSH-algebras 27 The last statement to be proved is that ~ irreducible primitive element in R~).

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