This package tests containment of (irreducible) varieties and computes Segre classes, algebraic multiplicity, and Fulton-MacPherson intersection products. More generally, for subschemes of ℙ^{n1}x...xℙ^{nm}, 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.

- 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