i1 : setRandomSeed 438; |
i2 : R = QQ[x,y,z,w] o2 = R o2 : PolynomialRing |
i3 : A = matrix{{x,y,z},{y,z,w}} o3 = | x y z | | y z w | 2 3 o3 : Matrix R <--- R |
i4 : chernClass minors(2,A) 3 2 o4 = 2H + 3H ZZ[H] o4 : ----- 4 H |
i5 : chernClass( minors(2,A), symbol t ) 3 2 o5 = 2t + 3t ZZ[t] o5 : ----- 4 t |
All the examples were done using symbolic computations with Gröbner bases. Changing the option Algorithm 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 ℙk. The Chern-Fulton class is one of several generalizations of Chern classes to possibly singular subschemes of projective space. It is defined as cCF(X) = c(Tℙk|X) ∩s(X,ℙk). 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.