Search results for "14M05"

showing 3 items of 3 documents

On the Betti numbers of three fat points in P1 × P1

2019

In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set Z of three fat points whose support is an almost complete intersection (ACI) in P1 × P1 . A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in P2 and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.

13F20Fat points Hilbert functions Multiprojective spaces13A15Fat pointsMathematics - Commutative Algebra13D40Mathematics - Algebraic GeometrySettore MAT/02 - AlgebraFat points; Hilbert functions; Multiprojective spacesMultiprojective spacesSettore MAT/03 - GeometriaMathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; Mathematics - Commutative Algebra; 13F20 13A15 13D40 14M0514M05Hilbert functions
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Multiprojective spaces and the arithmetically Cohen-Macaulay property

2019

AbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

Pure mathematicsArithmetically Cohen-Macaulay multiprojective spacesProperty (philosophy)points in multiprojective spaces arithmetically Cohen-Macaulay linkageGeneral MathematicsStar (graph theory)Space (mathematics)Commutative Algebra (math.AC)01 natural sciencesMathematics - Algebraic Geometryarithmetically Cohen-MacaulayTheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY0103 physical sciencesFOS: Mathematics0101 mathematicsAlgebraic Geometry (math.AG)Mathematics010102 general mathematics14M05 13C14 13C40 13H10 13A15Mathematics - Commutative Algebrapoints in multiprojective spacesAmbient spaceSettore MAT/02 - Algebra010307 mathematical physicsSettore MAT/03 - Geometrialinkage
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Expecting the unexpected: Quantifying the persistence of unexpected hypersurfaces

2021

If $X \subset \mathbb P^n$ is a reduced subscheme, we say that $X$ admits an unexpected hypersurface of degree $t$ for multiplicity $m$ if the imposition of having multiplicity $m$ at a general point $P$ fails to impose the expected number of conditions on the linear system of hypersurfaces of degree $t$ containing $X$. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understand. We introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of $X$ …

Pure mathematicsGeneral MathematicsComplete intersectionVector bundleAlgebraic geometrysymbols.namesakeMathematics - Algebraic GeometryAV-sequence; Complete intersection; Generic initial ideal; Hilbert function; Partial elimination ideal; Unexpected hypersurfaceUnexpected hypersurfaceFOS: MathematicsAlgebraic numberAV-sequenceAlgebraic Geometry (math.AG)Complete intersectionGeneric initial idealMathematicsHilbert series and Hilbert polynomialSequencePartial elimination idealSettore MAT/02 - AlgebraHypersurfaceHyperplanePrimary: 14C20 13D40 14Q10 14M10 Secondary: 14M05 14M07 13E10Hilbert functionsymbolsSettore MAT/03 - GeometriaAV-sequence Complete intersection Generic initial ideal Hilbert function Partial elimination ideal Unexpected hypersurface
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