Chart vs ambient coördinates for smooth affine varieties
In SageManifolds, coördinates are defined by means of charts and transition functions. For instance, I understand one can define a sphere by giving two charts and the usual transition arising from stereographic projections. But a sphere can also come with its own coordinates by being defined as a variety. Is there a way (or even a nice way!) to reconcile these two views when working with locally defined objects? For instance, I might have a function on a manifold that vanishes in some way, which when defined on charts is obvious, but when defined on the subvariety one needs to apply the defining equation(s) to see this. Other properties defined via equations in function rings (or other rings of sections of sheaves or bundles) are similar.
Either presentation includes information that the other doesn't: as you sketch, one can construct local charts from a projective model of a sphere, by gluing together charts obtained from projections. Choosing different projections will result in different charts, though. In the other direction the effect is even worse: you'll have to find a very ample line bundle, of which you can use the global sections to obtain a projective model, but the model you get depends on the line bundle you choose.
In short: there are mathematical problems with your question which you need to resolve before you can see what you can do in sage (and I don't think many of these things will be automated yet. Particularly representing and checking if line bundles are ample is likely not implemented yet)
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