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NormalToricVarieties :: kleinschmidt

kleinschmidt -- make a smooth toric variety with Picard rank two

Synopsis

Description

Peter Kleinschmidt constructs (up to isomorphism) all smooth normal toric varieties with dimension d and d+2 rays; see P. Kleinschmidt, A classification of toric varieties with few generators, Aequationes Mathematicae 35 (1998) 254-266.

When d=2, we obtain a variety isomorphic to a Hirzebruch surface.

i1 : X = kleinschmidt(2,{3});
i2 : rays X

o2 = {{-1, 0}, {1, 0}, {0, 1}, {3, -1}}

o2 : List
i3 : max X

o3 = {{0, 2}, {0, 3}, {1, 2}, {1, 3}}

o3 : List
i4 : FF3 = hirzebruchSurface 3;
i5 : rays FF3

o5 = {{1, 0}, {0, 1}, {-1, 3}, {0, -1}}

o5 : List
i6 : max FF3

o6 = {{0, 1}, {0, 3}, {1, 2}, {2, 3}}

o6 : List
The normal toric variety associated to the pair (d,A) is Fano if and only if ∑i=0r-1 ai < d-r+1.
i7 : X1 = kleinschmidt(3,{0,1});   
i8 : isFano X1

o8 = true
i9 : X2 = kleinschmidt(4,{0,0});   
i10 : isFano X2

o10 = true
i11 : ring X2

o11 = QQ[x , x , x , x , x , x ]
          0   1   2   3   4   5

o11 : PolynomialRing
i12 : X3 = kleinschmidt(9,{1,2,3}, CoefficientRing => ZZ/32003, Variable => y);
i13 : isFano X3

o13 = true
i14 : ring X3

        ZZ
o14 = -----[y , y , y , y , y , y , y , y , y , y , y  ]
      32003  0   1   2   3   4   5   6   7   8   9   10

o14 : PolynomialRing

See also

Ways to use kleinschmidt :