smoothFanoToricVariety(ZZ,ZZ) -- get a smooth Fano toric variety from database

Synopsis

Function: smoothFanoToricVariety
Usage:

smoothFanoToricVariety (d,i)
Inputs:
- d, an integer, equal to dimension of toric variety
- i, an integer, indexing a normal toric variety in database
Optional inputs:
- CoefficientRing => a ring, default value QQ, that specifies the coefficient ring of the total coordinate ring
- Variable => a symbol, default value x, that specifies the base name for the indexed variables in the total coordinate ring
Outputs:
- a normal toric variety, a smooth Fano toric variety

Description

This function accesses a database of all smooth Fano toric varieties of dimension at most $6$. The enumeration of the toric varieties follows Victor V. Batyrev's classification ( "On the classification of toric Fano", Journal of Mathematical Sciences (New York), 94 (1999) 1021-1050, arXiv:math/9801107v2 and Hiroshi Sato's "Toward the classification of higher-dimensional toric Fano varieties", The Tohoku Mathematical Journal. Second Series, 52 (2000) 383-413, arXiv:math/9011022) for dimension at most $4$ and Mikkel Øbro's classification ( "An algorithm for the classification of smooth Fano polytopes" arXiv:math/0704.0049v1) for dimensions $5$ and $6$.

There is a unique smooth Fano toric curve, five smooth Fano toric surfaces, eighteen smooth Fano toric threefolds, $124$ smooth Fano toric fourfolds, $866$ smooth Fano toric fivefolds, and $7622$ smooth Fano toric sixfolds.

For all $d$, smoothFanoToricVariety (d,0) yields projective $d$-space.

i1 : PP1 = smoothFanoToricVariety (1,0);

i2 : assert (rays PP1 === rays toricProjectiveSpace 1)

i3 : assert (max PP1 === max toricProjectiveSpace 1)

i4 : PP4 = smoothFanoToricVariety (4,0, CoefficientRing => ZZ/32003, Variable => y);

i5 : assert (rays PP4 === rays toricProjectiveSpace 4)

i6 : assert (max PP4 === max toricProjectiveSpace 4)

The following example was missing from Batyrev's table.

i7 : W = smoothFanoToricVariety (4,123);

i8 : rays W

o8 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {-1, -1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1},
     ------------------------------------------------------------------------
     {0, 0, -1, -1}, {1, 0, 1, 0}, {0, 1, 0, 1}, {-1, -1, -1, -1}}

o8 : List

i9 : max W

o9 = {{0, 1, 5, 6}, {0, 1, 5, 7}, {0, 1, 6, 7}, {0, 2, 4, 6}, {0, 2, 4, 8},
     ------------------------------------------------------------------------
     {0, 2, 6, 8}, {0, 4, 5, 7}, {0, 4, 5, 8}, {0, 4, 6, 7}, {0, 5, 6, 8},
     ------------------------------------------------------------------------
     {1, 2, 3, 7}, {1, 2, 3, 8}, {1, 2, 7, 8}, {1, 3, 5, 6}, {1, 3, 5, 8},
     ------------------------------------------------------------------------
     {1, 3, 6, 7}, {1, 5, 7, 8}, {2, 3, 4, 6}, {2, 3, 4, 7}, {2, 3, 6, 8},
     ------------------------------------------------------------------------
     {2, 4, 7, 8}, {3, 4, 6, 7}, {3, 5, 6, 8}, {4, 5, 7, 8}}

o9 : List

Acknowledgements

We thank Gavin Brown and Alexander Kasprzyk for their help extracting the data for the smooth Fano toric five and sixfolds from their Graded Rings Database.

Ways to use this method: