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--- status: draft
--- author(s): Decker, Popescu
--- notes:
document {
Key => genera,
Headline => "list of the successive linear sectional arithmetic genera",
SeeAlso => {genus,hilbertPolynomial,euler}
}
document {
Key => (genera,Ring),
Usage => "genera R",
Inputs => {"R"
},
Outputs => {List =>{"of the successive linear sectional
arithmetic genera of projective scheme ", TT "V"," with homogeneous coordinate
ring ", TT "R"}
},
"Computes the list of successive linear sectional arithmetic genera,
where the i-th entry in the list is the arithmetic genus of the i-th
successive generic hyperplane section of ", TT "V", " ",
"(= (-1)^dim-lin-section * (chi(OO_lin-section) - 1)).",
EXAMPLE {
"R = ZZ/101[x_0..x_4];",
"I = ideal random(R^1, R^{-2,-3});",
"genera(R/I)"
},
SeeAlso => {genus, euler}
}
document {
Key => {(genera,CoherentSheaf), (genera,Module)},
Usage => "genera M",
Inputs => {"M"
},
Outputs => {List =>{"of the successive linear sectional
arithmetic genera of the successive generic hyperplane restrictions
of ", TT "M"}
},
"Computes the list of successive generic linear sectional arithmetic genera,
where the i-th entry in the list is
(-1)^dim-support -i * (chi(M ** OO_lin-section) - 1)).",
EXAMPLE {
"V = Proj(ZZ/101[x_0..x_2]);",
"M = sheaf(image matrix {{x_0^3+x_1^3+x_2^3}})",
"genera M"
},
SeeAlso => {euler}
}
document {
Key => (genera,Ideal),
Usage => "genera I",
Inputs => {"I"
},
Outputs => {List =>{"of the successive linear sectional
arithmetic genera of ", TT "I"}
},
"Computes the list of successive linear sectional arithmetic genera,
where the i-th entry in the list is the arithmetic genus of the i-th
successive generic hyperplane section of the zero-locus of ", TT "I", " ",
"(= (-1)^dim-lin-section * (chi(OO_lin-section) - 1)).",
PARA{},
"A complete intersection of type (2,3) in projective fourspace;
its hyperplane section is a canonical curve of genus 4:",
EXAMPLE {
"R = ZZ/101[x_0..x_4];",
"I = ideal random(R^1, R^{-2,-3});",
"genera I"
},
SeeAlso => {euler, genus}
}
document {
Key => {(genera,ProjectiveVariety)},
Usage => "genera V",
Inputs => {"V"
},
Outputs => {List => {"of the successive linear sectional
arithmetic genera of ", TT "V"}
},
"Computes the list of successive linear sectional arithmetic genera,
where the i-th entry in the list is the arithmetic genus of the i-th
successive generic hyperplane section of ", TT "V", " ",
"(= (-1)^dim-lin-section * (chi(OO_lin-section) - 1)).",
PARA{},
"A complete intersection of type (2,3) in projective fourspace;
its hyperplane section is a canonical curve of genus 4:",
EXAMPLE {
"R = ZZ/101[x_0..x_4];",
"V = Proj(R/(ideal random(R^1, R^{-2,-3})));",
"genera V"
},
SeeAlso => {euler,genus}
}
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