1# gap> LoadPackage( "ToricVarieties" );
2# true
3# gap> sigma := Cone( [[ 15,-1 ],[0,1]] );
4# <A cone in |R^2>
5# gap> C15 := ToricVariety( sigma );
6# polymake: used package cddlib
7#   Implementation of the double description method of Motzkin et al.
8#   Copyright by Komei Fukuda.
9#   http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html
10#
11# <An affine normal toric variety of dimension 2>
12# gap> TorusInvariantDivisorGroup(C15);
13# <A free left module of rank 2 on free generators>
14# gap> ClassGroup(C15);
15# <A left module presented by 2 relations for 2 generators>
16# gap> Display(ByASmallerPresentation(last));
17# Z/< 15 >
18# gap> PicardGroup(C15);
19# <A zero left module>
20# gap> CartierTorusInvariantDivisorGroup(C15);
21# <A free left submodule given by 2 generators>
22# gap> Display(last);
23# [ [  15,   0 ],
24#   [  -1,   1 ] ]
25#
26# A left submodule generated by the 2 rows of the above matrix
27# gap> ImageSubobject(MapFromCharacterToPrincipalDivisor(C15));
28# <A free left submodule given by 2 generators>
29# gap> Display(last);
30# [ [  15,   0 ],
31#   [  -1,   1 ] ]
32#
33# A left submodule generated by the 2 rows of the above matrix
34# gap> IsSmooth(C15);
35# false
36
37
38LoadPackage( "ToricVarieties" );
39sigma := Cone( [[ 15,-1 ],[0,1]] );
40C15 := ToricVariety( sigma );
41TorusInvariantDivisorGroup(C15);
42ClassGroup(C15);
43Display(ByASmallerPresentation(last));
44PicardGroup(C15);
45CartierTorusInvariantDivisorGroup(C15);
46Display(last);
47ImageSubobject(MapFromCharacterToPrincipalDivisor(C15));
48Display(last);
49IsSmooth(C15);