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Currently I am reading "Les schemas de modules de courbes elliptiques" by Deligne and Rapoport and I got myself seriously confused about the following proposition (in English translation):


(II Proposition 1.15): Let $(p \colon C \to S, +)$ be a generalised elliptic curve. Then there exists a locally finite family of closed and disjoint subschemes $(S_n)_{n \ge 1}$ such that:

a) $\cup_{n} S_n = $ the image (under $p$) of the subscheme of non smothness $C^{sing}$.

b) Locally in the fppf topology over $S_n$ we have that $C$ is isomorphic to the pullback of the standard $n$-gon (from $Spec(\mathbb Z)$).


What troubles me is the closedness of the $S_n$. This seems to forbid that one can have curves of the following shape: Let $S$ be the Spec of one dimensional local ring. Let $C \to S$ be a generalised elliptic curve, such that the generic fibre is a rational nodal curve (a 1-gon) and special fibre is a 2-gon ($I_2$ in Kodaira Notation).

One can construct such a family by just starting with a constant family of 1-gons and then blowing up the crossing point in the special fibre. Or am I mistaken here?

In that case, $S_1$ should be the whole of $S$ and $S_2$ should be the closed point. But this contradicts the disjointness.

thanks in advance, Holger.

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4  
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). –  BCnrd Jul 24 '10 at 10:52
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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 –  BCnrd Jul 24 '10 at 11:04
    
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. –  Holger Partsch Jul 24 '10 at 11:50

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