Timeline for Kodaira-Spencer map as a "differential"

8 events

when toggle format	what		by	license	comment
Oct 16, 2012 at 16:58	history	edited	Choa	CC BY-SA 3.0	added 8 characters in body
Oct 16, 2012 at 16:44	comment	added	Matt		This can be made totally rigorous. There are a ton of details to check, but I think it is fairly straightforward when (as you point out) you know what the target of $f$ is supposed to be. Let $P$ be the Hilbert polynomial of $\mathcal{F}_x$. The image is $Hilb^P(X)$ (the scheme representing the functor $S\to X \mapsto $ flat quotients of $\mathcal{O}_{S\times X}$ with Hilbert polynomial $P$). Commutativity of the appropriate diagram and identification of tangent spaces with the Ext groups using deformation theory shows that $df$ is exactly $\kappa (x)$.
Oct 16, 2012 at 16:44	answer	added	Jonathan Wise		timeline score: 4
Oct 16, 2012 at 12:25	history	edited	Choa	CC BY-SA 3.0	improved formatting; added 11 characters in body
Oct 15, 2012 at 14:02	history	edited	Choa	CC BY-SA 3.0	added 5 characters in body
Oct 15, 2012 at 5:16	history	edited	Choa	CC BY-SA 3.0	added 209 characters in body; deleted 47 characters in body
Oct 15, 2012 at 1:14	history	edited	Choa	CC BY-SA 3.0	corrected spelling
Oct 14, 2012 at 15:58	history	asked	Choa	CC BY-SA 3.0