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The Defining Ideal of a Set of Points in Multi‐Projective Space
Author(s) -
van Tuyl Adam
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705006459
Subject(s) - ideal (ethics) , hilbert series and hilbert polynomial , conjecture , mathematics , position (finance) , set (abstract data type) , function (biology) , combinatorics , projective space , hilbert space , pure mathematics , discrete mathematics , projective test , computer science , philosophy , epistemology , finance , evolutionary biology , economics , biology , programming language
The defining ideal I X of a set of points X inℙ n 1× ... × ℙ n 1is investigated with a special emphasis on the case when X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, the degrees of the generators of the associated ideal I X are determined. ν( I X ) denotes the minimal number of generators of I X , and this description of the degrees is used to construct a function υ( s ; n 1 ,…, n k ) with the property that ν( I X )⩾ υ( s ; n 1 ,…, n k ) always holds for s points in generic position inℙ n 1× ... × ℙ n 1. When k = 1, υ( s ; n 1 ) equals the expected value for ν( I X ) as predicted by the ideal generation conjecture. If k ⩾ 2, it is shown that there are cases with ν( I X ) > υ( s ; n 1 , …, n k ). However, computational evidence suggests that in many cases ν( I X ) = υ( s ; n 1 , …, n k ).