Journal of the
Korean Mathematical Society JKMS

Consider a Noetherian domain $R_0$ with quotient field $K_0$. Let $K$ be a finitely generated regular transcendental field extension of $K_0$. We construct a Noetherian domain $R$ with $\mathrm{Quot}(R)=K$ that contains $R_0$ and embed $\mathrm{Spec}(R_0)$ into $\mathrm{Spec}$. Then, we prove that key properties of abelian varieties and smooth geometrically integral projective curves over $K$ are preserved under reduction modulo $\mathfrak{p}$ for ``almost all'' $\mathfrak{p}\in\mathrm{Spec}(R_0)$.

Keywords: Reduction, isotriviality, abelian variety, moduli space

MSC numbers: Primary 12E30, 14K05, 14H10