Semistable reduction theorem

In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism ${\displaystyle X\to S}$, there exists a morphism ${\displaystyle S'\to S}$ (called base change) such that ${\displaystyle X\times _{S}S'\to S'}$ is semistable (i.e., the singularities are mild in some sense). Precise formulations depend on the specific versions of the theorem. For example, if ${\displaystyle S}$ is the unit disk in ${\displaystyle \mathbb {C} }$, then "semistable" means that the special fiber is a divisor with normal crossings.[1]

The fundamental semistable reduction theorem for Abelian varieties by Grothendieck shows that if ${\displaystyle A}$ is an Abelian variety over the fraction field ${\displaystyle K}$ of a discrete valuation ring ${\displaystyle {\mathcal {O}}}$, then there is a finite field extension ${\displaystyle L/K}$ such that ${\displaystyle A_{(L)}=A\otimes _{K}L}$ has semistable reduction over the integral closure ${\displaystyle {\mathcal {O}}_{L}}$ of ${\displaystyle {\mathcal {O}}}$ in ${\displaystyle L}$. Semistability here means more precisely that if ${\displaystyle {\mathcal {A}}_{L}}$ is the Néron model of ${\displaystyle A_{(L)}}$ over ${\displaystyle {\mathcal {O}}_{L},}$ then the fibres ${\displaystyle {\mathcal {A}}_{L,s}}$ of ${\displaystyle {\mathcal {A}}_{L}}$ over the closed points ${\displaystyle s\in S=\mathrm {Spec} ({\mathcal {O}}_{L})}$ (which are always a smooth algebraic groups) are extensions of Abelian varieties by tori.[2] Here ${\displaystyle S}$ is the algebro-geometric analogue of "small" disc around the ${\displaystyle s\in S}$, and the condition of the theorem states essentially that ${\displaystyle A}$ can be thought of as a smooth family of Abelian varieties away from ${\displaystyle s}$; the conclusion then shows that after base change this "family" extends to the ${\displaystyle s}$ so that also the fibres over the ${\displaystyle s}$ are close to being Abelian varieties.

The imprortant semistable reduction theorem for algebraic curves was first proved by Deligne and Mumford.[3] 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.