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--- status: draft
--- author(s): Decker, Popescu
--- notes:
document {
Key => genus,
Headline => "arithmetic genus",
SeeAlso => {genera, euler}
}
document {
Key => {(genus,CoherentSheaf),(genus,Module),(genus, Ideal)},
Usage => "genus F",
Inputs => {"F" => {ofClass{CoherentSheaf,Module,Ideal} }},
Outputs => {ZZ },
"Computes the arithmetic genus of the coherent sheaf ", TT "F", " that is (-1)^dim-support * (chi(F) - 1)). If ", TT "F", " is a module
over a ring, then the genus of ", TT "F~", " is computed. If ", TT "I", " is an ideal in a ring ", TT "R", " then the genus of ", TT "(R/I)~", " is
computed.",
EXAMPLE {
"V = Proj(QQ[x,y,z]/ideal(y^2*z-x^2*(x+z)))",
"genus OO_V^1"
},
SeeAlso => {genera,euler}
}
document {
Key => (genus,ProjectiveVariety),
Usage => "genus V",
Inputs => {"V"
},
Outputs => {ZZ
},
"Computes the arithmetic genus of the projective scheme ", TT "V",
"A nodal plane cubic curve has arithmetic genus 1:",
EXAMPLE {
"V = Proj(QQ[a,b,c]/ideal(b^2*c-a^2*(a+c)))",
"genus V"
},
"The Fano model of a Reye type Enriques surface in projective fivespace:",
EXAMPLE {
"R = ZZ/101[x_0..x_5];",
"M = random(R^4, R^{4:-1});",
"I = minors(3, M+transpose(M));",
"V = Proj(R/I);",
"genus V"
},
SeeAlso => {genera,euler}
}
document {
Key => (genus,Ring),
Usage => "genus R",
Inputs => {"R"
},
Outputs => {ZZ
},
"Computes the arithmetic genus of the projective scheme ", TT "V",
" with homogeneous coordinate ring ", TT "R",
EXAMPLE {
"R = QQ[x,y,z]/ideal(y^2*z-x^3)",
"genus R"
},
SeeAlso => {genera,euler}
}
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