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We introduce the notion of pseudo-N\'eron model and give new examples ofvarieties admitting pseudo-N\'eron models other than Abelian varieties. As anapplication of pseudo-N\'eron models, given a scheme admitting a finitemorphism to an Abelian scheme over a positive-dimensional base, we prove thatfor a very general genus-0, degree-d curve in the base with d sufficientlylarge, every section of the scheme over the curve is contained in a uniquesection over the entire base.

Pseudo-N\'eron Model and Restriction of Sections with an appendix by Jason Michael Starr