Macaulay2 » Documentation
Packages » IncidenceCorrespondenceCohomology » incidenceCohomology
next | previous | forward | backward | up | index | toc

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

- Usage:
    `C = incidenceCohomology(L)`
    `C = incidenceCohomology(L, R)`
- Inputs:
  - L, a list, List of integers {i,p,a,b,n} List of integers {i,p,a,b,n} where i is the cohomological degree, p is the characteristic of the field, the couple (a,b) indicates the line bundle $\mathcal{O}_X(a,b)$ on the incidence correspondence $X\subseteq \mathbb{P}\times \mathbb{P}^\vee$, and $n$ is the dimension of $\mathbb{P} +1
  - R, a polynomial ring, Laurent polynomial ring, ambient ring for the character of the cohomology group, i.e. ZZ[x_1..x_n,x_1^{-1}..x_n^{-1}] where n-1 is the dimension of projective space
- Optional inputs:
  - FindCharacter => ..., default value false, An option to compute the character instead of the dimension
- Outputs:
  - C, an integer, dimension of the cohomology group (with the option to instead have the character)

## 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/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1740:0.