| Budget Amount *help |
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2002: ¥900,000 (Direct Cost: ¥900,000)
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| Research Abstract |
Let (a;n_1,n_2,...,n_a;n_{a+1},...,n_{a+b})(b>0) be the basic sequence of the saturated homogeneous ideal of an integral curve in a three dimensional projective space over an infinite field of characteristic zero.
(1)Suppose a>=2,b>=1. Let a' and t be integers with t+1<=a'<=a. If n_i=n_t+i-t for all i(t<=i<=a') and n_{a+1}<n_{a'}, then t(t-1)/2>max{i|n_{a+i}<n_{a'},1<=i<=b}.
(2)If there is an integer a'(1<=a'<=a) such that n_i=n_1+i-1 for all i(1<=i<=a'), then n_a'<=n_{a+1}.
(3)If there is an integer a'(1<=a'<=a) such that n_i=n_1+i-2 for all i(2<=i<=a'), then n_a'<=n_{a+1}.
The above three inequalities are valid for all curves which is contained in an irreducible reduced surface of degree a. We have also obtained the results below, whose details are omitted here.
(4)Necessary conditions that the basic sequence of an integral curve with b=1 must satisfy.
(5)The proof given by Decker-Schreyer of the Cook's connectedness conjecture under additional assumptions turned outs to be incomplete. We gave a new proof to that conjectur for a special case embracing the case set up by Decker-Schreyer's assumptions.
The sequences a,n_1,...,n_{a+b} satisfying all the above conditions and others known so far can be obtained by our computer program.
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