# When are Abelian schemes projective?

Under what conditions on the base $X$ are Abelian schemes $\mathcal{A}/X$ projective, and projective in which sense?

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Raynaud made an example over a 1-dimensional non-normal noetherian local domain with no ample line bundle. But if $X$ is normal and locally noetherian then $A$ admits an $X$-ample line bundle, so by using the 3rd tensor power, $A$ is closed in a twisted projective space bundle over $X$. Indeed, connected components of $X$ are open and irreducible, so a polarization $A_{\eta} \rightarrow A^{\vee}_{\eta}$ at a generic point $\eta\in X$ extends (by normality) to an isogeny $f$ over the entire connected component of $\eta$, and $(1,f)^{\ast}(P_{A/X})$ is $X$-ample: see Ch. 1 of Faltings-Chai. –  user29283 Apr 23 '13 at 14:11
Thank you very much! –  Timo Keller Apr 23 '13 at 14:22
When the base in integral and geom. unibranch, abelian schemes are projective; see Raynaud "Faisceaux amples..." XI.1.4. –  Kestutis Cesnavicius Apr 23 '13 at 16:04
Also thanks to you! –  Timo Keller Apr 23 '13 at 16:45