For an n-dimensional subscheme X of projective space ℙk, this command computes the push-forward of the total Chern-Schwartz-MacPherson class of X to the Chow ring of ℙk. The output is a polynomial in the hyperplane class, containing the degrees of the Chern-Schwartz-MacPherson classes (cSM)0(TX),...,(cSM)n(TX) as coefficients.
i1 : setRandomSeed 365;
|
i2 : R = ZZ/32749[x,y,z]
o2 = R
o2 : PolynomialRing
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i3 : CSMClass ideal(x^3 + x^2*z - y^2*z)
2
o3 = H + 3H
ZZ[H]
o3 : -----
3
H
|
i4 : chernClass ideal(x^3 + x^2*z - y^2*z)
o4 = 3H
ZZ[H]
o4 : -----
3
H
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We compute the Chern-Schwartz-MacPherson class of the singular cubic x
3 + x
2z = y
2z. Observe that it does not agree with the Chern-Fulton class computed by the command
chernClass. It is also possible to provide the symbol for the hyperplane class in the Chow ring of ℙ
k:
i5 : CSMClass( ideal(x^3 + x^2*z - y^2*z), symbol t )
2
o5 = t + 3t
ZZ[t]
o5 : -----
3
t
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All the examples were done using symbolic computations with Gröbner bases. The default algorithm computes the projective degrees using Gröbner bases. Changing the option Algorithm to ResidualSymbolic will compute the residual degrees using Gröbner bases. Changing the option Algorithm to Bertini will do the main computations numerically, provided Bertini is installed and configured .
Observe that the algorithm is a probabilistic algorithm and may give a wrong answer with a small but nonzero probability. Read more under
probabilistic algorithm.