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incidenceCohomology -- Computes dimension or character of sheaf cohomology of line bundles on the incidence correspondence

Description

This function computes the sheaf cohomology $H^i(X, \mathcal{O}_X(a,b))$, of the line bundle $\mathcal{O}_X(a,b)$ on the incidence correspondence $X$. The underlying characteristic of the field is p, prime or zero.

i1 : i = 0; p = 0; a = 7; b = 6; n = 3;
i6 : incidenceCohomology({i,p,a,b,n})

o6 = 420
i7 : i = 1; p = 3; a = 6; b = -8; n = 3;
i12 : incidenceCohomology({i,p,a,b,n})

o12 = 9
i13 : i = 1; p = 0; a = 6; b = -8; n = 3;
i18 : incidenceCohomology({i,p,a,b,n})

o18 = 0

We could instead ask for the character instead of just the dimension, setting the option FindCharacter to be true.

i19 : i = 3; p = 0; a = -2; b = -6; n = 3;
i24 : incidenceCohomology({i,p,a,b,n}, FindCharacter => true)

       4    3      3      2 2    2        2 2      3      2          2      3
o24 = x  + x x  + x x  + x x  + x x x  + x x  + x x  + x x x  + x x x  + x x 
       1    1 2    1 3    1 2    1 2 3    1 3    1 2    1 2 3    1 2 3    1 3
      -----------------------------------------------------------------------
         4    3      2 2      3    4
      + x  + x x  + x x  + x x  + x
         2    2 3    2 3    2 3    3

o24 : ZZ[x ..x ]
          1   3
i25 : i = 1; p = 3; a = 6; b = -8; n = 3;
i30 : incidenceCohomology({i,p,a,b,n}, FindCharacter => true)

       6 6 2    6 5 3    6 3 5    6 2 6    5 6 3    5 3 6    3 6 5    3 5 6  
o30 = x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       2 6 6
      x x x
       1 2 3

o30 : ZZ[x ..x ]
          1   3

Additional input of a polynomial ring allows the user to control the ambient ring of the character.

i31 : i = 2; p = 3; a = 7; b = -10; R = ZZ[x_1..x_3, Inverses=>true, MonomialOrder=>Lex];
i36 : incidenceCohomology({i,p,a,b},R)

       8 8      8 7 2    8 6 3    8 5 4    8 4 5    8 3 6    8 2 7    8   8  
o36 = x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       7 8 2    7 7 3    7 6 4    7 5 5    7 4 6    7 3 7    7 2 8    6 8 3  
      x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       6 7 4    6 6 5    6 5 6    6 4 7    6 3 8    5 8 4    5 7 5    5 6 6  
      x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       5 5 7    5 4 8    4 8 5    4 7 6    4 6 7    4 5 8    3 8 6    3 7 7  
      x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  + x x x  +
       1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3    1 2 3  
      -----------------------------------------------------------------------
       3 6 8    2 8 7    2 7 8      8 8
      x x x  + x x x  + x x x  + x x x
       1 2 3    1 2 3    1 2 3    1 2 3

o36 : R

Ways to use incidenceCohomology:

  • incidenceCohomology(List)
  • incidenceCohomology(List,PolynomialRing)

For the programmer

The object incidenceCohomology is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1740:0.