Timeline for Homogeneous polynomials cutting out complex abelian varieties
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jan 16 at 19:42 | vote | accept | Paul Cusson | ||
| Jan 16 at 10:59 | comment | added | Balazs | ...and then, somewhat accidentally, some of these abelian varieties will admit embeddings into smaller dimensional spaces by taking a subset of all sections (geometrically, a projection); but then the equations get more complicated. This must be the situation with the Horrocks-Mumford sections. | |
| Jan 16 at 10:58 | comment | added | Balazs | ...in the specific case of abelian varieties, I believe Mumford studies complete linear systems on an abelian variety A: you take an ample divisor L, and describe an embedding of A into projective space by taking all sections of L. Explicit sections are theta functions, and these satisfy the quadric and cubic relations Damian is talking about. Here we will have large codimension, and small degrees... | |
| Jan 16 at 10:54 | comment | added | Balazs | It is a basic result of a first projective geometry course that any projective variety can be described by (usually many) quadratic equations in a suitable (usually large) projective space. But this may be a very inefficient embedding, though possibly natural, depending on the situation. Just think of a hypersurface of some degree in projective space: it can be described by a single equation of large degree, or many equations of smaller degrees (=2). There is often a trade-off between the codimension and the degrees of the defining equations... | |
| Jan 16 at 7:42 | comment | added | Paul Cusson | In particular, I'm curious as how to interpret this comment by Damian in the linked MSE post: "The theory developped in the articles of D. Mumford, "On the equations defining abelian varieties I,II,II" in principle allows an explicit computation of equations cutting out abelian varieties in some projective spaces. In particular, he shows that quadratic and cubical equations suffice." | |
| Jan 16 at 7:41 | comment | added | Paul Cusson | Thank you for the answer, and sorry for the very late response. I guess I had assumed that any embedding of an abelian variety could be cut out by cubics. But from what you say, one should only see cubics when the embedding is in a much larger dimensional projective space, correct? I.e. one cannot have both a minimal dimension for the embedding AND have only cubics to cut out the locus? | |
| Nov 14, 2023 at 8:38 | history | edited | Balazs | CC BY-SA 4.0 |
added 161 characters in body
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| Nov 14, 2023 at 8:30 | history | answered | Balazs | CC BY-SA 4.0 |