2004 Fiscal Year Final Research Report Summary
Study of basic sequences of integral curves in three dimensional projective spaces
| Project/Area Number |
14540029
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
AMASAKI Mutsumi Hiroshima University, Graduate School of Education, Associate Professor, 大学院・教育学研究科, 助教授 (10243536)
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| Project Period (FY) |
2002 – 2004
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| Keywords | integral curve / basic sequence / connectedness / generic initial |
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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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