Macaulay2 / Macaulay2/M2

prune(CoherentSheaf) for NormalToricVarieties

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NormalToricVarieties
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Macaulay2
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Description

I had noticed this long ago but couldn't identify a simple enough example. The problem is that it seems like prune is not completely pruning on the first go:

needsPackage "NormalToricVarieties"
X = toricProjectiveSpace 2
S = ring X
M = coker matrix {{0,0,0,-x_0,-x_1,0,0,0,0,-x_0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,x_0,x_1,x_2,0,0,0,0,0,0,0,0,0,-x_0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,-x_0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{x_0,x_1,x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0,0,0,0,0,0,0,0,0,0,0},{-x_1,-x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0,0,0,0,0,0},{x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0},{0,0,0,0,0,0,0,x_0,x_1,x_2,0,0,0,x_0,x_1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,-x_0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,x_0,x_1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,x_0,x_1,x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_0,x_1,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_0,x_1,0,0,0,0,0},{0,0,0,0,0,x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_0,x_1},{0,0,0,0,0,0,0,0,0,0,0,-x_0,-x_1,-x_2,0,0,0,-x_0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,x_0,x_1,x_2,0,0,0,0,0,0,0,0,0,-x_0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0,-x_1,-x_2,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_0,-x_1,-x_2},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_0,x_1,x_2,0,0,0,x_0,x_1,x_2,0,0,0,0,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_1,-x_2,0,0,0,0,0,0,0,0,0,x_0,x_1,x_2,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_0,x_1,x_2,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-x_1,-x_2,0,0,0,-x_0,-x_1,-x_2,0,0,0,0,0,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_2,0,0,0,0,0,0,0,0,0,-x_0,-x_1,-x_2,0,0},{0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,x_2,0,0,0,0,x_1,x_2,0,0,0}}
prune sheaf M
prune oo

In the end we get:

i5 : prune sheaf M

o5 = cokernel {-4} | x_1  x_0  0    |
              {-4} | -x_2 0    x_0  |
              {-4} | 0    -x_2 -x_1 |

                                         3
o5 : coherent sheaf on X, quotient of OO  (4)
                                        X

i6 : prune oo

        1
o6 = OO  (5)
       X

o6 : coherent sheaf on X, free of rank 1

Note that same sheaf on a projective varieties defined from Varieties prunes fine.

Contributor guide

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First steps

  1. Read the whole issue, then the project's contributing guide.
  2. Comment on the issue to say you are picking it up — it saves two people doing the same work.
  3. Fork the repository and make your change on a branch.
  4. Open a pull request that references the issue number.

Research direction

Reproduce the NormalToricVarieties example using toricProjectiveSpace, the displayed matrix, and the two prune commands. Compare prune sheaf M with the subsequent prune oo result, then inspect the prune behavior for coherent sheaves on NormalToricVarieties versus Varieties. Done means the first pruning pass produces the expected fully pruned sheaf rather than requiring a second call.

Written by the indexing model from the issue text.

Assessment

Domain
tooling
Issue type
Bug
Difficulty
4/5
Estimated time
3-5 days
Activity status
Stale
Clarity
Needs clarification
Newbie friendliness
30/100

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