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NormalToricVarieties :: isNef

isNef -- whether a torus-invariant Weil divisor is nef

Synopsis

Description

A -Cartier divisor is nef (short for "numerically effective" or "numerically eventually free") if the intersection product of the divisor with every complete irreducible curve is nonnegative. The definition depends only on the numerical equivalence class of the divisor. For a torus-invariant-Cartier divisor D on a complete normal toric variety, the following are equivalent:
  • D is nef;
  • some positive integer multiply of D is Cartier and basepoint free;
  • the real piecewise linear support function associated to D is convex.
A torus-invariant Cartier divisor is nef if and only if it is basepoint free; in other words, the associated line bundle is generated by its global sections.

On a Hirzebruch surface, three of the four torus-invariant prime divisors are nef.

X1 = hirzebruchSurface 2;
isNef X1_0
isAmple X1_0
isNef X1_1
isNef X1_2
isAmple X1_2
isNef X1_3
isAmple X1_3
Not every -Cartier nef divisor is basepoint free.
X2 = weightedProjectiveSpace {2,3,5}
D = X2_1-X2_0
isNef D
HH^0(X2, OO D)
for i from 1 to dim X2 list HH^i(X2, OO D)
isCartier D
isCartier (30*D)
HH^0(X2, OO (30*D))
for i from 1 to dim X2 list HH^i(X2, OO (30*D))
There are smooth complete normal toric varieties with no nontrivial nef divisors.
R2 = {{1,0,0},{0,1,0},{0,0,1},{0,-1,2},{0,0,-1},{-1,1,-1},{-1,0,-1},{-1,-1,0}};
S2 = {{0,1,2},{0,2,3},{0,3,4},{0,4,5},{0,1,5},{1,2,7},{2,3,7},{3,4,7},{4,5,6},{4,6,7},{5,6,7},{1,5,7}};
X3 = normalToricVariety(R2,S2);
isComplete X3
isProjective X3
isSmooth X3
any(#rays X3, i -> isNef X3_i)
isNef (0*X3_1)
The most basic vanishing theorem for normal toric varieties states that the higher cohomology of coherent sheaf associated to a nef divisor is zero.
X4 = kleinschmidt(9,{1,2,3});
isNef X4_0
isAmple X4_0
for i from 1 to dim X4 list HH^i(X4, OO X4_0)
D = X4_0+X4_4
isNef D
isAmple D
for i from 1 to dim X4 list HH^i(X4, OO D)

See also

Ways to use isNef :