Development notes

Comparisons with other packages

This package is largely a translation of existing functionalities from Schubert2, Chow, and Schubert3 into Julia. Since most of the functionalities are already available elsewhere, the improvement in performance is important to justify its existence.

Schubert2

For Schubert2, we have already covered most of the functionalities, with the following exceptions:

• Isotropic flag bundles;
• Correspondence and IncidenceCorrespondence (what are the use cases for these?);
• Better support for Schubert classes.

Polynomial arithmetics in Macaulay2 are quite slow compared to Nemo and Singular. So most computations will automatically run a lot faster in this package than Schubert2.

Chow

For Chow, (almost?) all functionalities are covered, notably the construction of parameter spaces for twisted cubics, which is one of its key features. Moreover, we implemented constructors for moduli spaces of matrices with arbitrary size.

Most computations in this package should run faster with a constant factor of 2 to 5 when compared to Chow, thanks to the high efficiency of the Julia/Oscar combo. Note that the blowup procedure in Chow is particularly inefficient. See below for comparisons.

Schubert3

For Schubert3, we implemented the functionality of computing integrals using Bott's formula on Grassmannians, and on flag varieties in general. Moduli spaces of maps (Kontsevich's spaces) are not implemented yet.

Again, most computations in this package should run faster with a constant factor.

Others

A large part of the Singular library chern.lib is also covered.

Benchmarks

All the timings are done on an Intel(R) Core(TM) i5-8350U CPU @ 1.70GHz.

Computing Todd class

julia> @time todd(proj(200));
0.353537 seconds (989.07 k allocations: 44.766 MiB, 4.73% gc time)

Chow:

sage: time t = Proj(200).tangent_bundle().todd_class();
CPU times: user 1.23 s, sys: 715 µs, total: 1.23 s
Wall time: 1.19 s

Schubert3:

sage: time t = ProjectiveSpace(200).tangent_bundle().todd_class();
CPU times: user 2.15 s, sys: 1.15 ms, total: 2.15 s
Wall time: 2.15 s

Schubert2 and chern.lib are far slower so are not included.

Computing blowup

The computation of the blowup requires a "pushforward" function. In Macaulay2 there are two functions that do this: pushForward, and pushFwd from the PushForward package. pushFwd eventually calls pushForward so they are in fact the same.

Warning

Our implementation is still not optimal and is slower than Macaulay2's: the step using std to reduce the module is slow. Is this a Singular problem?

On the other hand, the counterpart in Chow, FiniteRingExtension, is very inefficient. This makes blowups in Chow very slow.

We compute the blowup of $\mathbf P^{20}$ at a point.

julia> @time blowup(point() → proj(20))[1] |> euler
0.011019 seconds (60.98 k allocations: 1.391 MiB)
40

Schubert2:

i12 : p = abstractProjectiveSpace 0; time euler first blowup map(abstractProjectiveSpace 20, p, OO_p)
-- used 1.09486 seconds

o13 = 40

Chow:

sage: time Blowup(PointChowScheme.hom("0", Proj(20))).codomain().euler_number()
CPU times: user 16.7 s, sys: 6.9 ms, total: 16.7 s
Wall time: 16.8 s
40

Schubert3 does not seem to provide this functionality.

Since the computations related to parameter spaces of twisted cubics use blowup, they run significantly faster in this package than with Chow.

• Compute the number of twisted cubics on a quintic threefold twisted_cubics_on_quintic_threefold: 5s v.s. 1min;
• Compute the Euler number of the LLSvS eightfold twisted_cubics_on_cubic_fourfold: 2min v.s. 20min.

Linear subspaces on hypersurfaces

Using Schubert calculus

julia> @time IntersectionTheory.linear_subspaces_on_hypersurface(1, 35, bott=false)
0.257564 seconds (1.66 M allocations: 30.382 MiB)
513687287764790207960329434065844597978401438841796875

julia> @time IntersectionTheory.linear_subspaces_on_hypersurface(2, 7, bott=false)
0.430082 seconds (2.68 M allocations: 42.083 MiB, 1.66% gc time)
279101475496912988004267637

Schubert2:

i4 : time integral chern symmetricPower_35 dual first bundles flagBundle{2,18}
-- used 15.2021 seconds

o4 = 513687287764790207960329434065844597978401438841796875

i5 : time integral chern symmetricPower_7 dual first bundles flagBundle{3,12}
-- used 12.3326 seconds

o5 = 279101475496912988004267637

Chow:

sage: time Grass(2,20).sheaves['universal_sub'].dual().symm(35).chern_classes()[-1].integral()
CPU times: user 787 ms, sys: 12.6 ms, total: 800 ms
Wall time: 812 ms
513687287764790207960329434065844597978401438841796875

sage: time Grass(3,15).sheaves['universal_sub'].dual().symm(7).chern_classes()[-1].integral()
CPU times: user 1.99 s, sys: 19.4 ms, total: 2.01 s
Wall time: 2.02 s
279101475496912988004267637

Schubert3 is much slower and is not included.

Using Bott's formula

julia> @time IntersectionTheory.linear_subspaces_on_hypersurface(1, 57)
0.263851 seconds (2.17 M allocations: 70.302 MiB, 36.51% gc time)
4543008461967180205447462399408892034555800983810817102531043233080341547902307166123135185210701405

julia> @time IntersectionTheory.linear_subspaces_on_hypersurface(3, 5)
0.546194 seconds (4.84 M allocations: 236.776 MiB, 31.41% gc time)
64127725294951805931404297113125

Schubert3 is slightly a bit faster when $k=1$, but slower with bigger $k$.

sage: time linear_subspaces_hypersurface(1,57,30)
CPU times: user 260 ms, sys: 0 ns, total: 260 ms
Wall time: 260 ms
4543008461967180205447462399408892034555800983810817102531043233080341547902307166123135185210701405

sage: time linear_subspaces_hypersurface(3,5,17)
CPU times: user 2.15 s, sys: 0 ns, total: 2.15 s
Wall time: 2.15 s
64127725294951805931404297113125

Some notes

Quotient rings

In a quotient ring, when should we do the reductions? It is obviously very costly to do this frequently; on the other hand, if we don't do it at all, we may accumulate some super complicated elements that will take a long time to reduce in the end. I find a good balance point is to do reduction for multiplications and powers, but not for additions and substractions. Here are some timings for a comparison. The bottleneck is when computing the Chern characters for the tautological subbundles using their Chern classes.

Current solution (reduction for * and ^ only)

julia> @time flag(3,6,9);
0.045371 seconds (489.13 k allocations: 6.366 MiB)

Reduction for all the operators

julia> @time flag(3,6,9);
0.161490 seconds (769.00 k allocations: 8.604 MiB)

No reduction at all

julia> @time flag(3,6,9);
1.168490 seconds (8.57 M allocations: 124.392 MiB, 1.07% gc time)

(Although this necessarily means that the ideal should be in a standard basis.)

Notes on Chow rings

• For a given AbsVariety, the underlying Singular ring of the Chow ring should never be modified: otherwise bundles already constructed will no longer be compatible. So whenever a new Singular ring is used, a new instance of AbsVariety should be created (e.g. _inclusion).

• On the other hand, the ideal could be changed but one should be careful. In particular, since basis_by_degree is cached (this is necessary due to performance reasons, since we need to check the number of zero cycles when doing integration), changing the ideal may lead to issues. But trim! is an exception since it does not change the structure and is in fact needed before performing basis_by_degree (otherwise the ideal might not be 0-dimensional and we won't be able to use kbase).