document and clean up algebraic cycles #4137

simonbrandhorst · 2024-09-23T16:09:29Z

No description provided.

simonbrandhorst · 2024-09-23T16:10:40Z

HechtiDerLachs

Looks good. Thank you for cleaning this up!

For now the mathematical description of the functions is spelled out in the documentation. We should, however, go through the literature and eventually adjust the notions and/or give references on the next appropriate occasion.

HechtiDerLachs · 2024-09-23T16:30:11Z

src/AlgebraicGeometry/Schemes/Divisors/Types.jl

+See [`trivializing_covering(C::EffectiveCartierDivisor)`](@ref) for how to
+The scheme $X$ is of type `CoveredSchemeType`.


This sentence seems to have been cut off.

HechtiDerLachs · 2024-09-24T07:24:45Z

I "fixed" the tests by coherently passing on the check argument so that the failure will hopefully not be triggered anymore. However, the problem remains for now, so I opened the issue #4138 .

simonbrandhorst · 2024-09-24T07:47:37Z

Thank you.

simonbrandhorst · 2024-09-24T07:50:52Z

src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl

-      temp_dict=IdDict{AbsAffineScheme,Ideal}()
-      temp_dict[U] = comp
-      I_sheaf_temp = IdealSheaf(X, extend!(cov, temp_dict), check=false)
+      I_sheaf_temp = PrimeIdealSheafFromChart(X, U, comp, check=false)


This is a non-trivial change.

I have some doubts about this function irreducible_decomposition(::EffectiveCartierDivisor).
It seems to assume that this Cartier divisor is a Weyl divisor. Do we need to assume the ambient scheme $X$ to be regular in codimension one for this to work? And if it is not, what is computed here?

Also I wonder if we could return a Weyl divisor here.

This is a non-trivial change.

This function was written long before we had PrimeIdealSheafFromChart. The change makes sense.

Also I wonder if we could return a Weyl divisor here.
Would make sense.

benlorenz

There are several calls to the now deprecated in_linear_system in /test/AlgebraicGeometry/Schemes/WeilDivisor.jl which now need the extra is_:

Oscar.jl/test/AlgebraicGeometry/Schemes/WeilDivisor.jl

Lines 126 to 130 in c35f32a

    
           @test in_linear_system(KK(x[1]), D) 
        
           @test !in_linear_system(KK(x[1]^2), D) 
        
           @test in_linear_system(KK(x[1]^2), 2*D) 
        
           # Not running at the moment; work in progress 
        
           #@test !in_linear_system(KK(x[1], x[2]), D)

(and further down as well)

HechtiDerLachs · 2024-09-24T12:24:44Z

@simonbrandhorst : Looks like we messed up our book tests again. Do you have an idea why?

…s, move some stuff

afkafkafk13

I leave you a few nitpicks -- mostly on the docstrings.

afkafkafk13 · 2024-09-24T19:30:20Z

src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl

+Return the `AlgebraicCycle` ``D = colength(I) ⋅ V(I)`` with coefficients 
+in ``R`` for an equidimensional sheaf of ideals ``I``.


The colength(I) here is confusing at first glance. Can you clarify by giving an appropriate example (doctest)?

afkafkafk13 · 2024-09-24T19:31:08Z

src/AlgebraicGeometry/Schemes/Divisors/AlgebraicCycles.jl

  D[I] = one(R)
  return D
 end

 @doc raw"""
    algebraic_cycle(I::AbsIdealSheaf, R::Ring) -> AlgebraicCycle

-Return the `AlgebraicCycle` ``D = 1 ⋅ V(I)`` with coefficients
-in ``R`` for a sheaf of prime ideals ``I``.
+Return the `AlgebraicCycle` defined by the equidimensional ideal sheaf `I`. 


This function is an alias for the other one, so it should be documented in the same way.

afkafkafk13 · 2024-09-24T19:41:41Z

src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl

-      temp_dict=IdDict{AbsAffineScheme,Ideal}()
-      temp_dict[U] = comp
-      I_sheaf_temp = IdealSheaf(X, extend!(cov, temp_dict), check=false)
+      I_sheaf_temp = PrimeIdealSheafFromChart(X, U, comp, check=false)


This is a non-trivial change.

This function was written long before we had PrimeIdealSheafFromChart. The change makes sense.

afkafkafk13 · 2024-09-24T19:42:13Z

src/AlgebraicGeometry/Schemes/Divisors/CartierDivisor.jl

-      temp_dict=IdDict{AbsAffineScheme,Ideal}()
-      temp_dict[U] = comp
-      I_sheaf_temp = IdealSheaf(X, extend!(cov, temp_dict), check=false)
+      I_sheaf_temp = PrimeIdealSheafFromChart(X, U, comp, check=false)


Also I wonder if we could return a Weyl divisor here.
Would make sense.

afkafkafk13 · 2024-09-24T19:45:59Z

src/AlgebraicGeometry/Schemes/Divisors/Types.jl

      for D in keys(coefficients)
-        is_equidimensional(D) || error("components of a divisor must be sheaves of equidimensional ideals")
+        (is_equidimensional(D) && dim(D) == d) || error("components of a cycle must be sheaves of equidimensional ideals")


You want to state "equidimensional of the same dimension" to stay consistent with the beginning of the line.

afkafkafk13 · 2024-09-24T19:47:40Z

src/AlgebraicGeometry/Schemes/Divisors/Types.jl

+@doc raw""" 
+    EffectiveCartierDivisor{CoveredSchemeType<:AbsCoveredScheme}
+
+An effective Cartier divisor on a scheme $X$ is a closed subscheme $D \subseteq S$ whose ideal sheaf $\mathcal{I}_D \subseteq \mathcal{O}_S$ is an invertible $\mathcal{O}_S$-module. In particular, $\mathcal{I}_D$ is locally principal.


Did something go wrong with X and S here?

afkafkafk13 · 2024-09-24T19:53:33Z

src/AlgebraicGeometry/Schemes/Divisors/WeilDivisor.jl

-  decomp = irreducible_decomposition(underlying_cycle(D))
-  return WeilDivisor(decomp, check=false)
+  E = irreducible_decomposition(underlying_cycle(D))
+  return WeilDivisor(E; check=false)


Why do you change the name of the variable? If irreducible_decomposition would return a divisor, I would agree with E here. But it is a list and therefore I personally find E misleading.

afkafkafk13 · 2024-09-24T19:59:23Z

src/AlgebraicGeometry/Schemes/Sheaves/IdealSheaves.jl

+@attr Bool function has_dimension_leq_zero(I::Ideal)
+  is_one(I) && return true
+  return dim(I) <= 0
+end
+
+@attr Bool function has_dimension_leq_zero(I::MPolyLocalizedIdeal)
+  R = base_ring(I)
+  P = base_ring(R)::MPolyRing
+  J = ideal(P, numerator.(gens(I)))
+  has_dimension_leq_zero(J) && return true
+  is_one(I) && return true
+  return dim(I) <= 0
+end
+
+@attr Bool function has_dimension_leq_zero(I::MPolyQuoLocalizedIdeal)
+  R = base_ring(I)
+  P = base_ring(R)::MPolyRing
+  J = ideal(P, lifted_numerator.(gens(I)))
+  has_dimension_leq_zero(J) && return true
+  is_one(I) && return true
+  return dim(I) <= 0
+end


A priori these functions do not have anything to do with IdealSheaves, they should rather find a place in src/Rings.

document algebraic cycles

49343cb

HechtiDerLachs approved these changes Sep 23, 2024

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HechtiDerLachs added 2 commits September 23, 2024 18:39

Fix some tests and replace if check by @check.

e4def6a

Pass on internal check argument.

c35f32a

HechtiDerLachs mentioned this pull request Sep 24, 2024

Equidimensional decomposition fails over extensions of IF_p #4138

Open

simonbrandhorst commented Sep 24, 2024

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benlorenz reviewed Sep 24, 2024

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fix iszero for cartier divisors, scheme to ambient_scheme for divisor…

9754647

…s, move some stuff

simonbrandhorst changed the title ~~document algebraic cycles~~ document and clean up algebraic cycles Sep 24, 2024

afkafkafk13 reviewed Sep 24, 2024

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document and clean up algebraic cycles #4137

document and clean up algebraic cycles #4137

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afkafkafk13 Sep 24, 2024

		See [`trivializing_covering(C::EffectiveCartierDivisor)`](@ref) for how to
		The scheme $X$ is of type `CoveredSchemeType`.

	@test in_linear_system(KK(x[1]), D)
	@test !in_linear_system(KK(x[1]^2), D)
	@test in_linear_system(KK(x[1]^2), 2*D)
	# Not running at the moment; work in progress
	#@test !in_linear_system(KK(x[1], x[2]), D)

		Return the `AlgebraicCycle` ``D = colength(I) ⋅ V(I)`` with coefficients
		in ``R`` for an equidimensional sheaf of ideals ``I``.

document and clean up algebraic cycles #4137

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document and clean up algebraic cycles #4137

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