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On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

2017

We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially ( P 1 ) n (\mathbb P^1)^n . A combinatorial characterization, the ( ⋆ ) (\star ) -property, is known in P 1 × P 1 \mathbb P^1 \times \mathbb P^1 . We propose a combinatorial property, ( ⋆ s ) (\star _s) with 2 ≤ s ≤ n 2\leq s\leq n , that directly generalizes the ( ⋆ ) (\star ) -property to ( P 1 ) n (\mathbb P^1)^n for larger n n . We show that X X is ACM if and only if it satisfies the ( ⋆ n ) (\star _n) -property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.

Property (philosophy)General MathematicsStar (game theory)Arithmetically Cohen-Macaulay; Linkage; Points in multiprojective spacescohen- macaulayCharacterization (mathematics)Commutative Algebra (math.AC)01 natural sciencesCombinatoricsMathematics - Algebraic GeometryPoints in multiprojective spaces0103 physical sciencesFOS: MathematicsProjective space0101 mathematicsFinite setAlgebraic Geometry (math.AG)multiprojective spacesMathematicsDiscrete mathematicsMathematics::Commutative AlgebraLinkageArithmetically Cohen-Macaulay Linkage Points in multiprojective spacesApplied Mathematics010102 general mathematicsExtension (predicate logic)Mathematics - Commutative AlgebraArithmetically Cohen-MacaulaypointsSettore MAT/02 - Algebracohen- macaulay multiprojective spaces points010307 mathematical physicsSettore MAT/03 - Geometria
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