mcolsson
commented
9 years ago Yes you are right, it is a bad reference. Here is a good one: XIII, 3.1 (b) in M. Raynaud, Faisceaux amples sur les sch ́emas en groupes et les espaces homog`enes, Springer Lecture Notes in Math 119, Springer-Verlag, Berlin (1970).
In Remark 8.4.15, it states that one has a failure of descent for families of genus 1 curves, citing Neron Models, section 6.7. But in Neron Models, the example seems to consist of taking a family of smooth genus 1 curves, blowing up a point in one of the fibers to obtain a fiber consisting of a normal crossings curve with components E, C with E genus 1 and C genus 0, and then contracting the curve E (the idea being that you can do this etale locally, but not Zariski locally over the base). This gives a family without effective descent, and so the total space is an algebraic space but not a scheme. On the other hand, it seems that this is not a family of genus 1 curves, since the changed fiber seems to consist of a single curve of genus 0. I suppose that perhaps this genus 0 curve will end up having a cusp, but in any case, the result doesn't seem to be a family of smooth genus 1 curves, and so doesn't seem to give an example of non-descent for smooth genus 1 curves...