Timeline for Curious propositon in "Les schemas de modules de courbes elliptiques"
Current License: CC BY-SA 2.5
4 events
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| Jul 24, 2010 at 11:50 | comment | added | Holger Partsch | thanks a lot, this is very helpful. Up to know I did not realize that the existence of a group scheme structure on smooth part is such a severe obstruction for a genus-1 curve. | |
| Jul 24, 2010 at 11:04 | comment | added | BCnrd | By the way, here is a direct argument (without reference to II, 1.15) that your proposed construction cannot work. If it did, then the smooth locus $C'$ in $C$ would admit a group scheme structure having the expected nature on fibers, and in particular $[2]:C' \rightarrow C'$ is a quasi-finite separated map which is flat (fibral flatness criterion), so $C'[2]$ is a quasi-finite flat separated group scheme. Its special fiber has rank 4 and generic fiber has rank 2. But Zariski's Main Theorem implies that for a q-finite flat sep'td map, fiber rank cannot grow under specialization. QED | |
| Jul 24, 2010 at 10:52 | comment | added | BCnrd | You didn't prove it has a structure of gen. elliptic curve (and it cannot, by II 1.15!); you just gave a curve whose geom. fibers are Neron polynons. A proper flat (finitely presented) curve with geom. fibers smooth genus 1 or Neron polygons may not admit a structure of gen. elliptic curve, even fpqc-locally on the base. See Rem. 2.1.13 of my paper "Arithmetic moduli of generalized elliptic curves" for an explicit example over an artin local ring (using a Fitting obstruction). D&R show that if geom. fibers are irred. then "geometry yields group theory", but false otherwise (e.g., 2-gons). | |
| Jul 24, 2010 at 10:14 | history | asked | Holger Partsch | CC BY-SA 2.5 |