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SegreClasses :: SegreClasses

SegreClasses -- Tests containment of varieties and computes algebraic multiplicity of subvarieties and Fulton-MacPherson intersection products - via a very general Segre class computation. The package works for subschemes of products of projective spaces.

Description

This package tests containment of (irreducible) varieties and computes Segre classes, algebraic multiplicity, and Fulton-MacPherson intersection products. More generally, for subschemes of ℙn1x...xℙnm, this package tests if a top-dimensional irreducible component of the scheme associated to an ideal is contained in the scheme associated to another ideal. Specialized methods to test the containment of a variety in the singular locus of another are provided, these methods work without computing the ideal of the singular locus and can provide significant speed-ups relative to the standard methods when the singular locus has a complicated structure. The package works for subschemes of products of projective spaces. The package implements methods described in [1]. More details and relevant definitions can be found in [1].

References:
[1] Corey Harris and Martin Helmer. "Segre class computation and practical applications." arXiv preprint arXiv:1806.07408 (2018). Link: https://arxiv.org/abs/1806.07408.

Authors

Version

This documentation describes version 1.02 of SegreClasses.

Source code

The source code from which this documentation is derived is in the file SegreClasses.m2.

Exports

  • Functions and commands
    • chowClass -- Finds the (fundamental) class of a subscheme in the Chow ring of the ambient space
    • containedInSingularLocus -- This method tests is an irreducible variety is contained in the singular locus of the reduced scheme of an irreducible scheme
    • intersectionProduct -- A class in the Chow ring of the ambient space representing the Fulton-MacPherson intersection product of two schemes inside a variety
    • isComponentContained -- This method tests containment of (irreducible) varieties; more generally it tests if a top-dimensional irreducible component of the scheme associated an ideal is contained in the scheme associated to another ideal
    • isMultiHom -- Tests if an ideal is multi-homogeneous with respect to the grading of its ring
    • makeChowRing -- Makes the Chow ring of a product of projective spaces.
    • makeProductRing -- Makes the coordinate ring of a product of projective spaces.
    • multiplicity -- This method computes the algebraic (Hilbert-Samuel) multiplicity
    • projectiveDegree -- This method computes a single projective degree of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces
    • projectiveDegrees -- This method computes the projective degrees of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces
    • segre -- This method computes the Segre class of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces
    • segreDimX -- This method computes the dimension X part of the Segre class of a scheme X inside a scheme Y, where X,Y are subschemes of some product of projective spaces