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Divisor :: divisorToModule

divisorToModule -- Calculate the corresponding module of a given divisor

Synopsis

Description

Get the associated module O(D) of a given Weil Divisor D.

i1 : R = QQ[x, y, u, v] / ideal(x * y - u * v)

o1 = R

o1 : QuotientRing
i2 : D1 = divisor({1, -2, 3, -4}, {ideal(x, u), ideal(x, v), ideal(y, u), ideal(y, v)})

o2 = 3*Div(y, u) + -4*Div(y, v) + 1*Div(x, u) + -2*Div(x, v) of R

o2 : WDiv
i3 : divisorToModule( D1 )

o3 = image {-2} | y4v   y3v2  y2v3  yv4   v5    |
           {-2} | y3uv  y2uv2 yuv3  uv4   xv4   |
           {-2} | y2u2v yu2v2 u2v3  xuv3  x2v3  |
           {-2} | yu3v  u3v2  xu2v2 x2uv2 x3v2  |
           {-2} | u4v   xu3v  x2u2v x3uv  x4v   |
           {-2} | xu4   x2u3  x3u2  x4u   x5    |
           {-2} | y5    y4v   y3v2  y2v3  yv4   |
           {-2} | y4u   y3uv  y2uv2 yuv3  uv4   |
           {-2} | y3u2  y2u2v yu2v2 u2v3  xuv3  |
           {-2} | y2u3  yu3v  u3v2  xu2v2 x2uv2 |
           {-2} | yu4   u4v   xu3v  x2u2v x3uv  |
           {-2} | u5    xu4   x2u3  x3u2  x4u   |

                             12
o3 : R-module, submodule of R

To get the associated module O(D) for a rational/real divisor D, we first obtain a new divisor D’ whose coefficients are the floor of the coefficients of D, and take O(D’) as O(D)

i4 : R = QQ[x, y, u, v] / ideal(x * y - u * v)

o4 = R

o4 : QuotientRing
i5 : D2 = divisor({3/5, -4/7, 9/4, -7/8}, {ideal(x, u), ideal(x, v), ideal(y, u), ideal(y, v)}, CoeffType=>QQ)

o5 = 9/4*Div(y, u) + -7/8*Div(y, v) + 3/5*Div(x, u) + -4/7*Div(x, v) of R

o5 : QDiv
i6 : divisorToModule( D2 )

o6 = image {-1} | y2 yv v2 |
           {-1} | u2 xu x2 |

                             2
o6 : R-module, submodule of R

See also

Ways to use divisorToModule :