In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism
(called base change) such that
is semistable (i.e., the singularities are mild in some sense).
Precise formulations depend on the specific versions of the theorem.
, 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
is an Abelian variety over the fraction field
of a discrete valuation ring
, then there is a finite field extension
has semistable reduction over the integral closure
(which are always a smooth algebraic groups) are extensions of Abelian varieties by tori.
is the algebro-geometric analogue of "small" disc around the
, and the condition of the theorem states essentially that
can be thought of as a smooth family of Abelian varieties away from
; the conclusion then shows that after base change this "family" extends to the
The important 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.