ToricVarieties: Goals in 2023 (and possibly beyond)

[HereAround]

Oscar-wide consolidation

• [ ] Canonical map (cf. https://github.com/oscar-system/Oscar.jl/issues/1095)
• [ ] Add @ref for is_trivial and is_irreducible (cf. https://github.com/oscar-system/Oscar.jl/issues/1471)

Spectral Sequences (likely also requires Oscar-wide consolidation for homological algebra)

• [ ] Add Spectral Sequence of Filtered Topological Space (cc @HechtiDerLachs ) (cf. CLs page 582 ff).

Intersection theory

• [ ] Use pre_inverse and post_inverse when applicable within the toric varieties code.
• [ ] There are various ring mappings involving the functionality of the Chow ring. Try to do the mappings more efficiently. For this we need a map from the Z^n graded Chow ring to the non-graded (or Z-graded) Chow ring.
• [ ] Intersect closed subvariety with rational equivalence class and represent the result as formal linear sum of closed subvariety of the given variety.
• [ ] Intersect two closed subvarieties and represent the result as formal linear sum of both subvarieties.
• [ ] Support coefficients of rational equivalence class, i.e. represent as formal linear sum of closed subvarieties (involving a choice) and return the corresponding coefficients.
• [ ] Along the same lines, provide function for the components of a representative of a rational equivalence class. Return these e.g. as closed subvarieties.

Integration of useful Singular libraries (schemes/varieties to affine toric varieties)

• [ ] toric.lib: Create toric ideal from exponent vectors (and compute a standard basis). This could be useful if we want to go from special schemes/varieties, namely those cut out by toric ideals to toric varieties. Benefits are to apply special toric methods (quicker, more powerful) than the generic scheme algorithms.
• [ ] cisimplicial.lib: Answers the question if a toric ideal a complete intersection.
• [ ] Possibly rewrite the existing C functionality in julia. (About 2000 lines of existing Singular code.)

Coherent sheaves (cf. https://github.com/oscar-system/Oscar.jl/pull/1809)

• [ ] Tangent and cotangent bundle of toric varieties.
• [ ] Normal and tangent bundle of complete intersections.
• [ ] Compute resolutions and use cohomCalg to try to compute their cohomologies. (Will of course not always work, but is a quick first check.)
• [ ] Extend to coherent sheaves and their cohomologies (this is the heart of the ToricVarieties_project).

Migration of other code of ToricVarieties_project

• [ ] QSMExplorer
• [ ] H0Approximator

Other

• [ ] Comparison among toric varieties (e.g. isprojective_space, isdirect_product_of_projective_space, isdelPezzo etc.).
• [ ] Construct toric varieties from a (proper) subset of all triangulations. In particular, add a method that constructs just one such variety. (This should eventually be available via https://github.com/Julia-meets-String-Theory/CYTools.jl).
• [ ] Use logical deductions?
• [ ] Toric blowups at arbitrary codimension. Provide constructor based on coordinate names that specify the blowup center. (https://github.com/oscar-system/Oscar.jl/pull/2228)
• [ ] Toric blow-down morphisms for blowups.
• [ ] Avoid prod for performance reasons.
• [ ] Toric blow-ups which allow do not require a smooth cone/cone contained in smooth cones only.

Completed

• [x] Integration of FtheoryTools (https://github.com/Julia-meets-String-Theory/FTheoryTools.jl) (https://github.com/oscar-system/Oscar.jl/pull/2092)
• [x] Toric schemes: More to come, but a beginning has been made (https://github.com/oscar-system/Oscar.jl/pull/1544).
• [x] Function that tells us the ray-coordinate-correspondence: https://github.com/oscar-system/Oscar.jl/pull/2245

[thofma]

How about using markdown syntax, e.g. # heading?

[HereAround]

How about using markdown syntax, e.g. # heading?

Done. Thx for the pointer.

[HechtiDerLachs]

• Tangent and cotangent bundle of toric varieties.

Have a look at #1840, if you like: There, I provide constructors for the (co-)tangent sheaves of arbitrary covered schemes. Through the work in #1636 this will also become available for toric varieties in the scheme theoretic sense.

However, I'm afraid that this is not precisely what you had in mind. I expect, you were thinking of these sheaves as something with a particular combinatorial description (like toric line bundles?). I would be interested in modeling a special type for these kind of sheaves which knows about their toric descriptions and build the bridge between the two sides. So that we can access different algorithmic backend for these, e.g. for cohomology computations, etc.