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 = QQ[x,y,z]
o2 = R
o2 : PolynomialRing
|
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
|
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
|
All the examples were done using symbolic computations with Gröbner bases. Changing the option ResidualStrategy 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.