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Cremona :: ChernSchwartzMacPherson

ChernSchwartzMacPherson -- Chern-Schwartz-MacPherson class of a projective scheme

Synopsis

Description

This is an example of application of the method projectiveDegrees, due to results shown in Computing characteristic classes of projective schemes, by P. Aluffi. See also the corresponding methods in the packages CSM-A, by P. Aluffi, and CharacteristicClasses, by M. Helmer and C. Jost.

In the example below, we compute the push-forward to the Chow ring of $\mathbb{P}^4$ of the Chern-Schwartz-MacPherson class of the cone over the twisted cubic curve, using both a probabilistic and a non-probabilistic approach.

i1 : GF(5^7)[x_0..x_4]

o1 = GF 78125[x ..x ]
               0   4

o1 : PolynomialRing
i2 : C = minors(2,matrix{{x_0,x_1,x_2},{x_1,x_2,x_3}})

               2                           2
o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
               1    0 2     1 2    0 3     2    1 3

o2 : Ideal of GF 78125[x ..x ]
                        0   4
i3 : time ChernSchwartzMacPherson C
     -- used 0.949315 seconds

       4     3     2
o3 = 3H  + 5H  + 3H

     ZZ[H]
o3 : -----
        5
       H
i4 : time ChernSchwartzMacPherson(C,MathMode=>true)
MathMode: output certified!
     -- used 1.194 seconds

       4     3     2
o4 = 3H  + 5H  + 3H

     ZZ[H]
o4 : -----
        5
       H
i5 : oo == ooo

o5 = true

In the case when the input ideal I defines a smooth projective variety $X$, the push-forward of $c_{SM}(X)$ can be computed much more efficiently using SegreClass. Indeed, in this case, $c_{SM}(X)$ coincides with the (total) Chern class of the tangent bundle of $X$ and can be obtained as follows (in general the method below gives the push-forward of the so-called Chern-Fulton class).

i6 : ChernClass = method(Options=>{MathMode=>false});
i7 : ChernClass (Ideal) := o -> (I) -> (
        s := SegreClass(I,MathMode=>o.MathMode);
        s*(1+first gens ring s)^(numgens ring I));
i8 : -- example: Chern class of G(1,4)
     G = Grassmannian(1,4,CoefficientRing=>ZZ/190181)

o8 = ideal (p   p    - p   p    + p   p   , p   p    - p   p    + p   p   ,
             2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4 
     ------------------------------------------------------------------------
     p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p   
      1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3
     ------------------------------------------------------------------------
     - p   p    + p   p   )
        0,2 1,3    0,1 2,3

                ZZ
o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
              190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
i9 : time ChernClass G
     -- used 0.127895 seconds

        9      8      7      6      5      4     3
o9 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H

     ZZ[H]
o9 : -----
       10
      H
i10 : time ChernClass(G,MathMode=>true)
MathMode: output certified!
     -- used 0.0463581 seconds

         9      8      7      6      5      4     3
o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H

      ZZ[H]
o10 : -----
        10
       H

See also

Ways to use ChernSchwartzMacPherson :

For the programmer

The object ChernSchwartzMacPherson is a method function with options.