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 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.

References

• Deligne, P.; Mumford, D. (1969). "The irreducibility of the space of curves of given genus". Publications Mathématiques de l'Institut des Hautes Scientifiques. 36 (36): 75–109. doi:10.1007/BF02684599. S2CID 16482150.
• Grothendieck, Alexandre (1972). Groupes de Monodromie en Géométrie Algébrique. Lecture Notes in Mathematics (in French). Vol. 288. Berlin; New York: Springer-Verlag. viii+523. doi:10.1007/BFb0068688. ISBN 978-3-540-05987-5. MR 0354656.
• Kempf, G.; Knudsen, Finn Faye; Mumford, David; Saint-Donat, B. (1973), Toroidal Embeddings I, Lecture Notes in Mathematics, vol. 339, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070318, ISBN 978-3-540-06432-9, MR 0335518
• Morrison, David R. (1984). "Chapter VI. The Clemens-Schmid exact sequence and applications" (PDF). Topics in Transcendental Algebraic Geometry. (AM-106). pp. 101–120. doi:10.1515/9781400881659-007. ISBN 9781400881659. S2CID 125739605.

Further reading

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