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CharacteristicClasses :: segreClass

segreClass -- Degrees of the Segre classes

Synopsis

Description

For an n-dimensional subscheme X of projective space Pk, this command computes the push-forward of the total Segre class s(X,Pk) of X in Pk to the Chow ring of Pk. The output is a polynomial in the hyperplane class, containing the degrees of the Segre classes s0(X,Pk),...,sn(X,Pk) as coefficients.
i1 : setRandomSeed 72;
i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing
i3 : segreClass ideal(x*y)

         2
o3 = - 4H  + 2H

     ZZ[H]
o3 : -----
        3
       H
i4 : segreClass ideal(x^2*y,x*y^2)

         2
o4 = - 3H  + 2H

     ZZ[H]
o4 : -----
        3
       H
We consider two singular curves in P2, C1 defined by {xy=0} and C2 defined by {x2y=xy2=0}. The degrees of their Segre classes are s0(C1,P2) = 2, s1(C1,P2)=-4, and s0(C2,P2)=2, s1(C2,P2)=-3. Observe that the two curves have the same underlying space but a different scheme structure, which is detected by the Segre classes. It is also possible to provide the symbol for the hyperplane class in the Chow ring of Pk:
i5 : segreClass( ideal(x*y), symbol t )

         2
o5 = - 4t  + 2t

     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 segreClass :