classWonderfulCompactification -- compute a toric cycle of X which class is the same as a given subvariety of X

Synopsis

Usage:

classWonderfulCompactification(X,I,J)
Inputs:
- X, a normal toric variety,
- I, an ideal,
- J, an ideal,
Outputs:
- D, a toric cycle,

Description

Given a hyperplane inside the torus, given by a linear ideal I, let X be a normal toric variety which fan is supported on the tropicalization of V(I). This function calculates a toric cycle which class is the same of a subvariety of V(I). The subvariety Y of V(I) is given by an ideal J such that Y = V(I+J), that is Y is the intersection of V(J) and V(I) inside the torus of X. The purpose of this function is essentially the same as classFromTropical, but it is optimized to this particular setting. Note that the closure of V(I) inside X is a wonderful compactification of V(I).

It is possible to explicitly input the toric variety, and this allows to implicitly specify the building set of the wonderful compactification.

A remarkable example among these is the moduli space of n-marked genus 0 curves M_0,n. Below, we use this function on M_0,6 to compute one of the 15 Keel-Vermeire divisors.

i1 : R = QQ[x_0..x_8];

i2 : I = ideal {-x_0+x_3+x_4, -x_1+x_3+x_5,-x_2+x_3+x_6, -x_0+x_2+x_7, -x_1+x_2+x_8, -x_0+x_1+1};

o2 : Ideal of R

i3 : X = normalToricVariety fan tropicalVariety I;

i4 : f = x_0*x_1-x_2*x_3;

i5 : D = classWonderfulCompactification(X,I,f)

o5 = 2*X    + 2*X     - X     + 2*X     + 2*X     - X     + X    - X    - 2*X    + X
        {9}      {10}    {11}      {13}      {14}    {17}    {2}    {5}      {6}    {7}

o5 : ToricCycle on X

Ways to use classWonderfulCompactification:

classWonderfulCompactification(Ideal,Ideal)
classWonderfulCompactification(Ideal,RingElement)
classWonderfulCompactification(NormalToricVariety,Ideal,Ideal)
classWonderfulCompactification(NormalToricVariety,Ideal,RingElement)

For the programmer

The object classWonderfulCompactification is a method function.