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
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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
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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
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i6 : divisorToModule( D2 )
o6 = image {-1} | y2 yv v2 |
{-1} | u2 xu x2 |
2
o6 : R-module, submodule of R
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