Timeline for Surjectivity of map of Picard schemes implies abelian
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2017 at 3:36 | comment | added | rollover | @nfdc23 Thank you! It all makes sense now. :) Feel free to post this as an answer so I can accept it. | |
| Nov 11, 2017 at 17:24 | comment | added | nfdc23 | The identity component of the Picard scheme of a projective scheme $Z$ over a field $k$ is generally not proper, even for a singular integral curve (e.g., for $Z$ the nodal cubic it is $\mathbf{G}_m$); the real content is that ${\rm{Pic}}^0_{D/k}$ is proper (given that a-priori it is smooth since $\dim D = 1$), and this is what we get from the surjectivity of the map from ${\rm{Pic}}^0_{X/k}$ that is already known to be proper. | |
| Nov 11, 2017 at 12:52 | comment | added | rollover | If $Pic^0$ of any proper 1-dim. scheme is smooth, what do you make of the formulation "we see that $Pic^0X \rightarrow Pic^0D$ is surjective, and because $Pic^0X$ is an Abelian variety, we conclude that $Pic^0D$ is also an Abelian variety"? If $Pic^0D$ is smooth and therefore Abelian anyways, then this seems quite redundant.. | |
| Nov 11, 2017 at 12:51 | comment | added | rollover | @nfdc23 thank you for your swift comment! I saw this (I believe) in Mumford's Lecture notes on curves on algebraic surfaces, lecture 27, as indicated in both the paper and Badescu's book. I currently forget the exact formulation, and don't have access to either M's lectures or the book you mentioned until monday. Do you happen to have the exact statement? | |
| Nov 11, 2017 at 5:07 | comment | added | nfdc23 | The Picard scheme of a proper 1-dimensional scheme is always smooth since the infinitesimal criterion is a consequence of the vanishing of coherent ${\rm{H}}^2$ on such "curves" (see 8.4/2 in the book Neron Models). So the target ${\rm{Pic}}^0_{D/k}$ is a smooth connected group variety, and inherits properness by surjectivity, whence it is an abelian variety. | |
| Nov 11, 2017 at 3:32 | history | asked | rollover | CC BY-SA 3.0 |