chernClass -- computes degrees of the Chern classes

Synopsis

Usage:
chernClass I
chernClass X
Inputs:
- I, an ideal, a homogeneous ideal in a polynomial ring over a field, defining a projective scheme X
- X, a projective variety
Optional inputs:
- ResidualStrategy => Symbolic, default value Symbolic, the strategy to compute the degrees of the residuals
Outputs:
- a ring element, the pushforward of the total Chern class of the scheme X to the Chow ring ZZ[H]/(H^k+1) of projective space P^k.

Description

For a non-singular n-dimensional subscheme X of projective space P^k, this command computes the push-forward of the total Chern class of X to the Chow ring of P^k. The output is a polynomial in the hyperplane class, containing the degrees of the Chern classes c₀(T_X),...,c_n(T_X) as coefficients.

i1 : R = QQ[x,y,z,w]

o1 = R

o1 : PolynomialRing

i2 : A = matrix{{x,y,z},{y,z,w}}

o2 = | x y z |
     | y z w |

             2       3
o2 : Matrix R  <--- R

i3 : chernClass minors(2,A)

       3     2
o3 = 2H  + 3H

     ZZ[H]
o3 : -----
        4
       H

The 2x2-minors of the matrix A form the ideal of the twisted cubic. It is well-known that its degree is 3 and its genus is 0. The calculations confirm that deg c₁ = 2-2g = 2 and deg c₀ = 3. It is also possible to provide the symbol for the hyperplane class in the Chow ring of P^k:

i4 : chernClass( minors(2,A), symbol t )

       3     2
o4 = 2t  + 3t

     ZZ[t]
o4 : -----
        4
       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 .

The command chernClass actually computes the push-forward of the total Chern-Fulton class of the subscheme X of projective space P^k. The Chern-Fulton class is one of several generalizations of Chern classes to possibly singular subschemes of projective space. It is defined as c_CF(X) = c(T_P^k|_X) ∩s(X,P^k). For non-singular schemes, the Chern-Fulton class coincides with the Chern class of the tangent bundle. So for non-singular input, the command will compute just the usual Chern class.

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 `chernClass` :

chernClass(Ideal)
chernClass(Ideal,Symbol)
chernClass(ProjectiveVariety)
chernClass(ProjectiveVariety,Symbol)

chernClass -- computes degrees of the Chern classes

Synopsis

Description

Ways to use chernClass :

Ways to use `chernClass` :