Timeline for Elegant definition for the scheme parametrizing $g_d^r$'s on a curve
Current License: CC BY-SA 3.0
22 events
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| Dec 15, 2013 at 16:30 | comment | added | meh | @ abramo - I think the comments of Smith and Huizenga explicate my point. | |
| Dec 14, 2013 at 18:44 | comment | added | Abramo | Thanks a lot @roysmith, I will read it as soon as possible! | |
| Dec 14, 2013 at 18:26 | comment | added | roy smith | If you do not have access to Kempf's book, one of his seminars is reproduced in the appendix to this paper: math.uga.edu/%7Eroy/onparam.pdf. Geometrically the key idea there (apparently due to Mumford) is that by adding a base divisor, one embeds the fibers of varying dimension of a low degree Abel map, inside the family of constant dimension fibers of a high degree Abel map. Then one can trivialize the latter family locally, obtaining a representation of the previous fibers as subspaces of a fixed projective space, with determinantal equations for the image in the Jacobian. | |
| Dec 13, 2013 at 19:51 | comment | added | roy smith | One can almost dispense with the base divisor in the case of special divisors which one may assume to have form K-D. Then H^(K-D) is the kernel and H^1(K-D) is (one dimension higher than) the cokernel of the map H^0(K)-->H^0(K|D). This is the sheaf version of Roch's evaluation map, dual to Riemann's "period map" H^0(O(D)|D)-->H^1(O), from the point of view taken in the Contemp. Math. article. Here however the divisor D will vary with K-D. | |
| Dec 13, 2013 at 17:39 | comment | added | roy smith | I think Jack hits the key issue. An elementary take on the base divisor D is to produce a sheaf sequence 0-->L-->L(D)-->L(D)|D-->0, such that H^0(L) and H^1(L) are simply kernel and cokernel of the associated restriction map H^0(L(D))-->H^0(L(D)|D). This allows one to model spaces of sections of line bundles nearby L as kernels of matrices of the same dimension. A detailed discussion of the role of base divisors is in Kempf's Abelian Integrals, (or lacking that, some related comments are in Contemp. Math. vol 397, p.215 ff. where I admit we also tried at first to avoid them.) | |
| Dec 13, 2013 at 14:19 | comment | added | Jack Huizenga | In particular, it is certainly not the case that the fiber of the push forward of the Poincare bundle at some line bundle is the global sections of the bundle. | |
| Dec 13, 2013 at 14:06 | comment | added | Jack Huizenga | It seems to me that in many cases of interest the push forward of L will be 0 or perhaps locally free of rank smaller than r+1. The spaces of sections of special linear series don't extend to nearby line bundles, so the push forward of L cannot be of any use. Twisting by a sufficiently ample line bundle kills cohomology so that pushforwards are well behaved; when cohomology jumps uncontrollably the push forward is not well behaved. | |
| Dec 13, 2013 at 12:54 | history | edited | Abramo | CC BY-SA 3.0 |
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| Dec 13, 2013 at 12:33 | comment | added | Abramo | @roysmith: Could you please give it a look when you have the time? | |
| Dec 13, 2013 at 12:32 | history | edited | Abramo | CC BY-SA 3.0 |
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| Dec 13, 2013 at 11:47 | comment | added | Abramo | @roysmith: The main problem in the definition of $G_d^r$ in GAC is that it (apparently) depends on a choice of a high degree divisor $\Gamma$. A posteriori one can show that this definition is independent of the choice made. Regarding $W_d^r$, one fixes this "issue" using Fitting. It would be nice to do something similar for $G_d^r$ as well, avoiding to choose a divisor $\Gamma$. I will update my answer soon to make this point clear | |
| Dec 12, 2013 at 22:09 | comment | added | roy smith | I am trying to understand. Take r=0. There the variety of 0-dim linear series is the symmetric product of the curve, which can indeed be given a scheme structure without noticing it is also a natural "blowup" of the variety of line bundles. Noticing that fact however, seems to me the modern contribution to the theory, since modeling the Abel map on the "kernel resolution" of the space of singular matrices, is what allowed Mumford -Kempf to improve Riemann's results. But I admit the determinant structure on Theta, historically came after that on the symmetric product. Is this your idea? | |
| Dec 12, 2013 at 18:29 | comment | added | Abramo | @aginensky: What do you mean by "your G(L) varies with L"? My $G(\mathscr{L})$ is defined in terms of the Poincaré line bundle, and it doesn't depend on a specific $L\in Pic^d$. | |
| Dec 12, 2013 at 18:26 | history | edited | Abramo | CC BY-SA 3.0 |
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| Dec 12, 2013 at 18:19 | comment | added | Abramo | @roysmith: In those pages they define the scheme structure in a way which is not elegant from my point of view. I would like to achieve the same with a fiber product that doesn't involve the generic determinantal variety $\tilde{M}_k$ | |
| Dec 12, 2013 at 18:16 | comment | added | meh | i didn't vote, but I would think that there would be no reason to write anything more than what is in GAC. I agree with abx, and many others, that it is 'nicely done' there. Finally, I would add that your $G(L)$ varies with $L$ and only seems to make sense in the first place on some $G^r_d$. I'm not sure it really is well defined. | |
| Dec 12, 2013 at 18:11 | comment | added | roy smith | It is not clear to me if you want a different way of stating the scheme structure (which on p.83-84 does look like a fiber product to me), or a different result for the scheme structure. It seems the prop. 3.6 p.184 of ACGH implies they have the "correct" scheme structure. | |
| Dec 12, 2013 at 17:50 | comment | added | Abramo | I got one downvote. Could the downvoter explain his/her reasons? This way I may be able to improve my question | |
| Dec 12, 2013 at 17:15 | comment | added | Abramo | @abx: I know that book, but I'm not satisfied with the way they define it there. I'm sure there's a better, more modern approach to it | |
| Dec 12, 2013 at 17:09 | comment | added | abx | All of this is nicely done in "Geometry of Algebraic Curves" (I), by Arbarello et al. | |
| Dec 12, 2013 at 16:19 | history | edited | Abramo | CC BY-SA 3.0 |
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| Dec 12, 2013 at 16:13 | history | asked | Abramo | CC BY-SA 3.0 |