semistable reduction theorem

{{Short description|Mathematical theory in the field of algebraic geometry}}

In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism of schemes $X\; \backslash to\; S$, there exists a morphism $S\text{'}\; \backslash to\; S$ (called base change) such that $X\; \backslash times\_S\; S\text{'}\; \backslash to\; S\text{'}$ is semistable (i.e., the singularities are mild in some sense). Precise formulations depend on the specific versions of the theorem.

For example, if $S$ is the unit disk in $\backslash mathbb\{C\}$, then "semistable" means that the special fiber is a divisor with normal crossings.{{harvnb|Morrison|1984|loc=§ 1.}}

The fundamental semistable reduction theorem for Abelian varieties by Grothendieck shows that if $A$ is an Abelian variety over the fraction field $K$ of a discrete valuation ring $\backslash mathcal\{O\}$, then there is a finite field extension $L/K$ such that $A\_\{(L)\}\; =\; A\; \backslash otimes\_K\; L$ has semistable reduction over the integral closure $\backslash mathcal\{O\}\_L$ of $\backslash mathcal\{O\}$ in $L$. Semistability here means more precisely that if $\backslash mathcal\{A\}\_L$ is the Néron model of $A\_\{(L)\}$ over $\backslash mathcal\{O\}\_L,$ then the fibres $\backslash mathcal\{A\}\_\{L,s\}$ of $\backslash mathcal\{A\}\_L$ over the closed points $s\backslash in\; S=\backslash mathrm\{Spec\}(\backslash mathcal\{O\}\_L)$ (which are always a smooth algebraic groups) are extensions of Abelian varieties by tori.Grothendieck (1972), Théorème 3.6, p. 351

Here $S$ is the algebro-geometric analogue of "small" disc around the $s\backslash in\; S$, and the condition of the theorem states essentially that $A$ can be thought of as a smooth family of Abelian varieties away from $s$; the conclusion then shows that after base change this "family" extends to the $s$ so that also the fibres over the $s$ are close to being Abelian varieties.

The important semistable reduction theorem for algebraic curves was first proved by Deligne and Mumford.{{harvnb|Deligne|Mumford|1969|loc=Corollary 2.7.}} The proof proceeds by showing that the curve has semistable reduction if and only if its Jacobian variety (which is an Abelian variety) has semistable reduction; one then applies the theorem for Abelian varieties above.