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

splittingPrincipalParts -- Computes the torus equivariant splitting type of the bundle of principal parts on $\mathbb{P}^1$

Description

This function computes the splitting type as a direct sum of line bundles of the k-th bundle of principal parts for $\mathcal{O}(m)$ on $\mathbb{P}^1$ over a field of characteristic p using the relationship between $\mathcal{P}^k(\mathcal{O}(m))$ and the bundle $\mathcal{F}^d_r$ as described in [KMRR,24].

The user inputs the prime p and the integers m and k.

The output is a list whose entries are either integers if Multidegree is false, or lists of three integers if Multidegree is true. When Multidegree is false, the integers in the list are the twists for the line bundles occurring in the decomposition. When Multidegree is true, for each element of the list, the first entry is the corresponding twist for the line bundle, and the second and third entries are the multidegrees arising from a torus action upon choosing coordinates for $\mathbb{P}^1$.

i1 : p = 5;
i2 : m = 10; k = 7;
i4 : splittingPrincipalParts(p,m,k)

o4 = {5, 2, 2, 5, 2, 2, 3, 3}

o4 : List
i5 : p = 0;
i6 : m = 10; k = 7;
i8 : splittingPrincipalParts(p,m,k)

o8 = {3, 3, 3, 3, 3, 3, 3, 3}

o8 : List

We compute the same example but now we also compute the multidegrees.

i9 : p = 5;
i10 : m = 10; r= 7;
i12 : splittingPrincipalParts(p,m,k, Multidegree=>true)

o12 = {{5, 0, 5}, {2, 1, 7}, {2, 2, 6}, {5, 5, 0}, {2, 6, 2}, {2, 7, 1}, {3,
      -----------------------------------------------------------------------
      3, 4}, {3, 4, 3}}

o12 : List

Ways to use splittingPrincipalParts:

  • splittingPrincipalParts(ZZ,ZZ,ZZ)

For the programmer

The object splittingPrincipalParts is a method function with options.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/IncidenceCorrespondenceCohomology.m2:1323:0.