Timeline for Kodaira-Spencer map in a concrete instance
Current License: CC BY-SA 2.5
12 events
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| Mar 20, 2019 at 16:36 | comment | added | Pulcinella | It might be worth noting that for smooth $X$, its first order deformations are $H^1(X,\mathcal{T}_X)$ follows immediately after you know that all deformations of affine $X$ are trivial, and automorphisms of $X\times_kk[\epsilon]$ over (affine) $X$ are precisely vector fields on $X$. For the latter, if $A$ is a $k$-algebra, a $k$-linear derivation (i.e. vector field) $D:A\to A$ corresponds to the automorphism of $A\otimes_kk[\epsilon]$ fixing $\epsilon$ and sending $a\mapsto a+ \epsilon D(a)$. | |
| May 13, 2010 at 20:39 | comment | added | Qfwfq | @jlk: yes, I think what you say can be a right interpretation of my question. | |
| May 13, 2010 at 14:41 | comment | added | jlk | (continued): I think I understand your question better now. The hyperelliptic curve admits a degree 2 map to P^1, and deforming this map is closely related to deforming its branch locus (i.e. the Weierstrass points). Every deformation of the map to P^1 induces a deformation of X; you'd like to see these deformations written down in such a way that you can compare them to the deformations of the Weierstrass points. Is that a reasonable interpretation of your question? If so, I think this can be done. I am traveling, but I can try to write something down when I get back next week. | |
| May 13, 2010 at 14:38 | comment | added | jlk | @unknown: By "deforming the equation" I meant deforming a curve X on a surface S by deforming it as a closed subscheme (i.e. finding a k[e]-flat closed subscheme in X \times k[e] that is equal to X when e=0). If you write everything out, then these deformations are constructed by deforming the equations of X in S. There is a natural map H^{0}(O(X)|){X}) \to H^{1}(O), and the image of this is the deformations of X that can be realized as embedded deformations. However, I don't think this is what you are interested in. | |
| May 13, 2010 at 10:38 | comment | added | Qfwfq | (continued) Also, could you please explain what do you mean by "you can deform the curve by deforming the equation"? Isn't it that in this specific case [as every h.ell. curve can be obtained as a branched cover of P^1 via the above equation] all such deformations are obtained by "deforming the equation" (in that specific way)? | |
| May 13, 2010 at 10:35 | comment | added | Qfwfq | @jlk: I was interested in a 1-parameter first-order deformation arising moving the Weierstrass points of the curve, hence (I thought) deforming the (desingularized curve corresponding to the) above equation. I don't know if this is related to embedded deformations of curves inside projective surfaces; if you think yes, it'd be interesting to see the details. | |
| May 13, 2010 at 5:30 | comment | added | jlk | @unknown, given a curve X on a smooth, projective surface S, you can deform the curve by deforming the equation of the curve and these deformations are computed by a map H^{0}(X,O_{X}(X)) \to H^1(X,O). After typing the answer above, I realized this might be closer to your original question. Let me know if you are interested in the details | |
| May 13, 2010 at 3:47 | history | edited | jlk | CC BY-SA 2.5 |
edited body
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| May 9, 2010 at 4:03 | history | edited | jlk | CC BY-SA 2.5 |
Genus was wrong.
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| May 9, 2010 at 3:50 | history | edited | jlk | CC BY-SA 2.5 |
added 1588 characters in body; added 239 characters in body
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| May 9, 2010 at 1:11 | history | edited | jlk | CC BY-SA 2.5 |
Another exponent issue fixed.; edited body
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| May 9, 2010 at 1:06 | history | answered | jlk | CC BY-SA 2.5 |