Semistable reduction theorem
In algebraic geometry, semistable reduction theorems state that, given a proper flat morphism , there exists a 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. For example, if  is the unit disk in , then "semistable" means that the special fiber is a divisor with normal crossings. 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  such that  has semistable reduction over the integral closure  of  in . Semistability here means more precisely that if  is the Néron model of  over  then the fibres  of  over the closed points  (which are always a smooth algebraic groups) are extensions of Abelian varieties by tori. Here  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  so that also the fibres over the  are close to being Abelian varieties. The imprortant semistable reduction theorem for algebraic curves was first proved by Deligne and Mumford. 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. References

Further reading 
The Stacks Project Chapter 55: Semistable Reduction: Introduction, https://stacks.math.columbia.edu/tag/0C2Q

Algebraic geometry