John Carlos Baez on Nostr: nprofile1q…zgt5c "could you elaborate on how projective geometry appears here?" I ...
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"could you elaborate on how projective geometry appears here?"
I wouldn't say "projective geometry", I'd say "projective algebraic geometry": the study of projective varieties, which are certain nice subspaces of projective space:
https://en.wikipedia.org/wiki/Projective_variety
It's a very long story, but I summarized it in my post (4/n).
An affine variety is just another way of talking about a commutative ring, namely the ring of regular functions on that variety. For a projective variety this fails, since there are usually very few regular functions on a projective variety.
This is why algebraic geometers turned to working with sheaves in the 1940s. They often study a projective variety by looking at the quasicoherent sheaves on that variety. These form a 2-rig, and this 2-rig has all the information you need.
Jim's idea is to simply work directly with the 2-rig. In many videos he's been developing algebraic geometry from this viewpoint.
"could you elaborate on how projective geometry appears here?"
I wouldn't say "projective geometry", I'd say "projective algebraic geometry": the study of projective varieties, which are certain nice subspaces of projective space:
https://en.wikipedia.org/wiki/Projective_variety
It's a very long story, but I summarized it in my post (4/n).
An affine variety is just another way of talking about a commutative ring, namely the ring of regular functions on that variety. For a projective variety this fails, since there are usually very few regular functions on a projective variety.
This is why algebraic geometers turned to working with sheaves in the 1940s. They often study a projective variety by looking at the quasicoherent sheaves on that variety. These form a 2-rig, and this 2-rig has all the information you need.
Jim's idea is to simply work directly with the 2-rig. In many videos he's been developing algebraic geometry from this viewpoint.