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--- status: draft
--- author(s): Sorin Popescu
--- notes:
document {
Key => euler,
Headline => "Euler characteristic",
SeeAlso => {eulers, genus}
}
document {
Key => (euler,ProjectiveHilbertPolynomial),
Headline => "constant term of the Hilbert polynomial",
Usage => "euler P",
Inputs => {"P"
},
Outputs => {ZZ =>" the constant term of the Hilbert polynomial"},
"The command returns ", TT "P(0)", " the constant term of P.
This is also the Euler characteristic of the sheaf of rings of a projective
variety with Hilbert polynomial ", TT "P", ".",
PARA{},
EXAMPLE {
"R = QQ[x_0..x_3]",
"C = Proj(R/monomialCurveIdeal(R, {1,3,4}));",
"P = hilbertPolynomial C",
"euler P"
},
SeeAlso => {hilbertPolynomial,eulers,genus}
}
document {
Key => (euler,ProjectiveVariety),
Headline => "topological Euler characteristic of a (smooth) projective variety",
Usage => "euler V",
Inputs => {"V"
},
Outputs => {ZZ =>"the topological Euler characteristics of the variety V"
},
"The command computes the topological Euler characteristic of the (smooth) projective
variety V as an alternated sum of its Hodge numbers. The Hodge numbers can be computed
directly using the command ", TO "hh", ".",
PARA{},
"A smooth plane quartic curve has genus 3 and topological Euler characteristic -4:",
EXAMPLE {
"Quartic = Proj(QQ[x_0..x_2]/ideal(x_0^4+x_1^4+x_2^4))",
"euler(Quartic)"
},
PARA{},
"The topological Euler characteristic of a smooth quintic hypersurface in
projective fourspace is -200:",
EXAMPLE {
"Quintic = Proj(QQ[x_0..x_4]/ideal(x_0^5+x_1^5+x_2^5+x_3^5+x_4^5-101*x_0*x_1*x_2*x_3*x_4))",
"euler(Quintic)"
},
Caveat => {"No test is made to see if the projective variety is smooth"},
SeeAlso => {Proj,genus,hh}
}
document {
Key => (euler,CoherentSheaf),
Headline => "Euler characteristic of coherent sheaf",
Usage => "euler F",
Inputs => {"F"
},
Outputs => {ZZ =>" the Euler characteristic of the cohomology of the sheaf"},
"The command returns ", TT "chi(F)", " the Euler characteristic of the sheaf ",
TT "F", " i.e. the alternated sum of the dimensions of its cohomology groups",
PARA{},
SeeAlso => {(eulers,CoherentSheaf),genus}
}
document {
Key => (euler,Module),
"This needs to be documented.",
SeeAlso => {(eulers,Module),genus}
}
document {
Key => (euler,Ring),
"This needs to be documented.",
SeeAlso => {(eulers,Ring),genus}
}
document {
Key => (euler,Ideal),
"This needs to be documented.",
SeeAlso => {(eulers,Ideal),genus}
}
document {
Key => eulers,
Headline => "list the sectional Euler characteristics",
SeeAlso => {euler,genera,genus}
}
document {
Key => {(eulers, CoherentSheaf),(eulers,Module)},
Usage => "eulers E",
Inputs => {"E"
},
Outputs => {List =>"the successive sectional Euler characteristics of a coherent sheaf, or a module."
},
"Computes a list of the successive sectional Euler characteristics of a coherent sheaf,
the i-th entry on the list being the Euler characteristic of the i-th
generic hyperplane restriction of ", TT "E",
PARA{},
"The Horrocks-Mumford bundle on the projective fourspace:",
EXAMPLE {
"R = QQ[x_0..x_4];",
"a = {1,0,0,0,0}",
"b = {0,1,0,0,1}",
"c = {0,0,1,1,0}",
"M1 = matrix table(5,5, (i,j)-> x_((i+j)%5)*a_((i-j)%5))",
"M2 = matrix table(5,5, (i,j)-> x_((i+j)%5)*b_((i-j)%5))",
"M3 = matrix table(5,5, (i,j)-> x_((i+j)%5)*c_((i-j)%5))",
"M = M1 | M2 | M3;",
"betti (C=res coker M)",
"N = transpose submatrix(C.dd_3,{10..28},{2..36});",
"betti (D=res coker N)",
"Pfour = Proj(R)",
"HorrocksMumford = sheaf(coker D.dd_3);",
"HH^0(HorrocksMumford(1))",
"HH^0(HorrocksMumford(2))",
"eulers(HorrocksMumford(2))"
},
SeeAlso => {genera,genus}
}
document {
Key => (eulers,Ideal),
Usage => "eulers I",
Inputs => {"I"
},
Outputs => {List =>"the successive sectional Euler
characteristics of an ideal (sheaf)."
},
"Computes a list of the successive sectional Euler
characteristics of an ideal (sheaf), the i-th entry
in the list being the Euler characteristic of the i-th
generic hyperplane restriction of ", TT "I",
EXAMPLE {
"R = ZZ/101[a,b,c];",
"I =ideal(a^3+b^3+c^3)",
"eulers I"
},
SeeAlso => {genera,genus}
}
document {
Key => (eulers,Ring),
Usage => "eulers R",
Inputs => {"R"
},
Outputs => {List =>"the successive sectional Euler
characteristics of a (sheaf of) ring(s)."
},
"Computes a list of the successive sectional Euler
characteristics of a ring (sheaf of), the i-th entry
in the list being the Euler characteristic of the i-th
generic hyperplane restriction of ", TT "R",
EXAMPLE {
"S = ZZ/101[a,b,c];",
"I = ideal(a^3+b^3+c^3)",
"R = S/I",
"eulers(R)",
"J = substitute(ideal(b,a+c),R)",
"eulers(R/J)"
},
SeeAlso => {genera,genus},
}
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