## Cartier test for toric divisors on non-full-dimensional cones

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justus-springer
Posts: 2
Joined: 01 Feb 2023, 11:12

### Cartier test for toric divisors on non-full-dimensional cones

Hello,

I am new to Polymake and I'm trying to use it for computations in toric geometry.

I have tried the following example to test if a toric divisor is Cartier, but got an error that I don't understand:

Code: Select all

polytope > application "fulton"; fulton > $tv = new NormalToricVariety(new PolyhedralFan(INPUT_RAYS=>[[-1,-1,0],[-1,-1,-2],[3,0,2],[0,3,2]], INPUT_CONES=>[[0,1],[0,2,3],[1,2,3]])); fulton > print$tv->PROJECTIVE; polymake: used package cdd cddlib Implementation of the double description method of Motzkin et al. Copyright by Komei Fukuda. http://www-oldurls.inf.ethz.ch/personal/fukudak/cdd_home/ false fulton > $td =$tv->DIVISOR(COEFFICIENTS=>[1,1,1,1]); fulton > print $td->Q_CARTIER; polymake: WARNING: rule DIVISOR.Q_CARTIER, DIVISOR.CARTIER_DATA : RAYS, DIVISOR.COEFFICIENTS, MAXIMAL_CONES, LINEAL ITY_SPACE ( applied to parent ) failed: too many solutions for Cartier data true fulton > print$td->CARTIER; polymake: WARNING: could not compute 'CARTIER' probably because of unsatisfied preconditions: precondition : PRINCIPAL ( CARTIER : ) precondition : SMOOTH_FAN ( DIVISOR.CARTIER : ) ( applied to parent ) precondition : DIVISOR.Q_CARTIER ( DIVISOR.CARTIER : ) ( applied to parent ) 
I am entering a normal non-complete toric variety with four rays in three dimensions. There are two three-dimensional cones and one two-dimensional. I assume the error at "print \$td->CARTIER" has something to do with the warning in the line earlier (?) "Too many solutions for Cartier data". If I enter a toric variety where all the maximal cones are of maximal dimension, the error disappears.

Are these types of toric varieties (non-complete, some maximal cones are not of maximal dimension) not fully supported in polymake?

I'm grateful for any explanation.

lkastner
Developer
Posts: 12
Joined: 27 May 2012, 23:35

### Re: Cartier test for toric divisors on non-full-dimensional cones

That is correct, we use 4.2.8 of Cox, Little, Schenck, and did not deal with (1) yet.

justus-springer
Posts: 2
Joined: 01 Feb 2023, 11:12

### Re: Cartier test for toric divisors on non-full-dimensional cones

Thanks for the info!