TropicalToric -- Macaulay2 package for toric intersection theory using tropical geometry.

Description

This package implements the computations of classes in the Chow ring of a toric variety using tropical geometry. Part of this code is based on a package previously written by Diane Maclagan, Sameera Vemulapalli, Corey Harris, Erika Pirnes and Ritvik Ramkumar.

Author

Alessio Borzì <[email protected]>

Certification

Version 1.0 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2023-09-14, in the article Tropical computations for toric intersection theory in Macaulay2 (DOI: 10.2140/jsag.2024.14.19). That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 1.0 of TropicalToric.

Source code

The source code from which this documentation is derived is in the file TropicalToric.m2. The auxiliary files accompanying it are in the directory TropicalToric/.

Exports

Types
- ToricCycle -- the class of a toric cycle on a NormalToricVariety
Functions and commands
- classFromTropical -- compute a toric cycle of X which class is the same as a given subvariety of X
- classFromTropicalCox -- compute a toric cycle of X which class is the same as a given subvariety of X
- classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X
- degCycle -- Compute the degree of a cycle of maximal codimension
- isTransverse -- checks transversality of toric divisors and cones
- makeTransverse -- Find a divisor linearly equivalent to a toric divisor D that is transverse to a given cone C
- poincareDuality -- Apply the isomorphism Hom(A^k(X),ZZ) = A_k(X)
- poincareMatrix -- see poincareDuality -- Apply the isomorphism Hom(A^k(X),ZZ) = A_k(X)
- polymakeConeContains -- Check if a vector is contained in a cone using polymake
- pushforwardMultiplicity -- pushforward of Minkowski weights
- refineMultiplicity -- Compute the multiplicities of a refinement of a tropical variety
- toricCycle -- Creates a ToricCycle
- toricDivisorFromCycle -- Convert a codimension one toric cycle into a toric divisor
- torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus
Methods
- classFromTropical(NormalToricVariety,Ideal) -- see classFromTropical -- compute a toric cycle of X which class is the same as a given subvariety of X
- classFromTropicalCox(NormalToricVariety,Ideal) -- see classFromTropicalCox -- compute a toric cycle of X which class is the same as a given subvariety of X
- classWonderfulCompactification(Ideal,Ideal) -- see classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X
- classWonderfulCompactification(Ideal,RingElement) -- see classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X
- classWonderfulCompactification(NormalToricVariety,Ideal,Ideal) -- see classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X
- classWonderfulCompactification(NormalToricVariety,Ideal,RingElement) -- see classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X
- degCycle(ToricCycle) -- see degCycle -- Compute the degree of a cycle of maximal codimension
- isTransverse(ToricDivisor,List) -- see isTransverse -- checks transversality of toric divisors and cones
- makeTransverse(ToricDivisor,List) -- see makeTransverse -- Find a divisor linearly equivalent to a toric divisor D that is transverse to a given cone C
- NormalToricVariety _ List -- Create toric cycles
- poincareDuality(List,NormalToricVariety,ZZ) -- see poincareDuality -- Apply the isomorphism Hom(A^k(X),ZZ) = A_k(X)
- poincareMatrix(NormalToricVariety,ZZ) -- see poincareDuality -- Apply the isomorphism Hom(A^k(X),ZZ) = A_k(X)
- polymakeConeContains(List,List) -- see polymakeConeContains -- Check if a vector is contained in a cone using polymake
- pushforwardMultiplicity(NormalToricVariety,NormalToricVariety,List,ZZ) -- see pushforwardMultiplicity -- pushforward of Minkowski weights
- refineMultiplicity(List,Fan,NormalToricVariety) -- see refineMultiplicity -- Compute the multiplicities of a refinement of a tropical variety
- refineMultiplicity(TropicalCycle,NormalToricVariety) -- see refineMultiplicity -- Compute the multiplicities of a refinement of a tropical variety
- support(ToricCycle) -- Get the list of cones with non-zero coefficients in the cycle
- expression(ToricCycle) -- see ToricCycle -- the class of a toric cycle on a NormalToricVariety
- net(ToricCycle) -- see ToricCycle -- the class of a toric cycle on a NormalToricVariety
- toricCycle(List,List,NormalToricVariety) -- see toricCycle -- Creates a ToricCycle
- toricCycle(List,NormalToricVariety) -- see toricCycle -- Creates a ToricCycle
- ToricCycle * ToricCycle -- intersection product of two toric cycles
- - ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- QQ * ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- QQ * ToricDivisor -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricCycle + ToricDivisor -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricCycle - ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricCycle - ToricDivisor -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricDivisor + ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricDivisor - ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ZZ * ToricCycle -- see ToricCycle + ToricCycle -- perform arithmetic on toric cycles
- ToricCycle == ToricCycle -- equality of toric cycles
- ToricCycle _ List -- Coefficients of toric cycle
- ToricCycle _ ZZ -- see ToricCycle _ List -- Coefficients of toric cycle
- toricCycle(ToricCycle) -- see toricCycle(ToricDivisor) -- Creates a ToricCycle from a ToricDivisor
- toricCycle(ToricDivisor) -- Creates a ToricCycle from a ToricDivisor
- ToricDivisor * List -- restriction of a Cartier toric divisor to the orbit closure of a cone
- ToricCycle * ToricDivisor -- see ToricDivisor * ToricCycle -- intersection product of a ToricDivisor and ToricCycle
- ToricDivisor * ToricCycle -- intersection product of a ToricDivisor and ToricCycle
- toricDivisorFromCycle(ToricCycle) -- see toricDivisorFromCycle -- Convert a codimension one toric cycle into a toric divisor
- torusIntersection(NormalToricVariety,Ideal) -- see torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus
- torusIntersection(NormalToricVariety,Ideal,Ring) -- see torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus
- torusIntersection(NormalToricVariety,RingElement) -- see torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus
- torusIntersection(NormalToricVariety,RingElement,Ring) -- see torusIntersection -- compute the ideal of the intersection of a subvariety of a toric variety with the torus
- variety(ToricCycle) -- Get the ambient variety of the cycle
Symbols
- PoincareMatrix -- see poincareDuality -- Apply the isomorphism Hom(A^k(X),ZZ) = A_k(X)

For the programmer

The object TropicalToric is a package.