complexity of `orbits` and hence `toricBlowup` is exponential for blowing up at a point
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Description
I noticed that `toricBlowup` for a point in `PP^n` seems to grow exponentially: ~1s, ~3s, ~10s, ~60s for n=14,15,16,17. Try this with:
```m2
n = 17; elapsedTime toricBlowup(toList(0..n-1), toricProjectiveSpace n)
```
The main bottleneck seems to be computation of the orbits.
I haven't read the implementation carefully, but even if the implemented algorithm is optimal for the general case, there are a few potential improvements here for a student or workshop:
1. Both `toricBlowup` and `orbits` could be improved using mutable lists and hash tables.
2. Blowups at points should be trivial to compute using a simpler algorithm.
3. The basis for the Picard group chosen for these special blowups could also be chosen better. In #4014 I added a WeilToClass option to specify my own degree map, but ideally this should be automatic.
cc: @ggsmith
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Research direction
Start by running the `toricBlowup(toList(0..n-1), toricProjectiveSpace n)` example and reading the implementations of `toricBlowup` and `orbits`, which the issue identifies as the bottleneck. Determine which improvement is in scope and compare runtime and resulting blowups for point centers before and after the change.
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Assessment
- Domain
- performance
- Issue type
- Bug
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Needs clarification
- Newbie friendliness
- 30/100