Decomposition info #2504
Decomposition info #2504
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| P3 = projective_space(QQ, 3) | ||
| X = covered_scheme(P3) | ||
| S = homogeneous_coordinate_ring(P3) | ||
| (x,y, z, w) = gens(S) | ||
| I = ideal(S, [x^3+y^3+z^3+w^3, x+y+z+w, 37*x^2-x*z+4*w^2-5*w*z]) | ||
| II = ideal_sheaf(P3, I) | ||
| @test oscar.colength(II) == 6 | ||
| J = ideal(S, [x, y, z^4]) | ||
| JJ = ideal_sheaf(P3, J) | ||
| @test oscar.colength(JJ) == 4 |
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These are the relevant tests for this PR.
| # The elements in h indicate where components must | ||
| # be located so that they can not be spotted in other charts. | ||
| # We iteratively single out these components by adding a sufficiently high | ||
| # power of the equation to the ideal. |
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...and this here how this can be brought to work.
The next step would be to automatically keep track of this information for blowups. Then integration of cycles should go through in little time once we allow weil divisor components to be non-prime.
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We also need this information to down to subschemes.
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If you agree with this concept, I could move forward with it. |
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May I sum up the idea, how I understood it from the first glance (i5 minutes after receiving the first mail on this PR): |
Codecov Report
Additional details and impacted files@@ Coverage Diff @@
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- Coverage 72.95% 72.95% -0.01%
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One general remark: Make sure to correctly carry along this information through a blow up and hence through any sequence of blow-ups. |
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From my side this is good to go. |
| of elements ``fᵢ₁,…,fᵢᵣ ∈ 𝒪(Uᵢ)``. These elements are chosen so that for every | ||
| affine patch `Uᵢ` of ``X`` in the covering `C` the closed subvarieties ``Zᵢ ⊂ Uᵢ`` | ||
| defined by the ``fᵢⱼ`` give rise to a decomposition of ``X`` as a **disjoint** union | ||
| ``X = \bigcup_{i} Z_i`` of locally closed subvarieties. |
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I still don't understand this.
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What is
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decomposition_info(X)[U_i].
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O.k. please add this to the docstring with the next pr.
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But still how do you get
Shouldn't
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This must hold by construction of the decomposition_info. You do not get a decomposition with inclusion maps, etc. You get a list of ring elements for every patch and you may continue working with them while assuming that the decomposition described above is a disjoint union.
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Still don't get it. Because I do not get your definition of a decomposition_info.
Please give clean mathematical definition.
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I set to auto merge. Please improve the docstring with the next pr. |
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@HechtiDerLachs promised to add the check later. So I merge this now. |
We introduce a new pattern to keep track of a decomposition of a covered scheme as a disjoint union. The extended version of this should allow for the computation of intersection numbers without computing any glueings.
Let me know, what you think @simonbrandhorst @afkafkafk13 .