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CharacteristicClasses :: CSMClass

CSMClass -- computes degrees of the Chern-Schwartz-MacPherson classes

Synopsis

Description

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 x3 + x2z = y2z. 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 Pk:
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.

Ways to use CSMClass :