Semistable reduction theorem

In algebraic geometry, the semistable reduction theorem states that, given a proper flat morphism $X\to S$, there exists a morphism $S'\to S$ (called base change) such that $X\times _{S}S'\to S'$ is semistable (i.e., the singularities are mild in some sense).

For example, if $S$ is the unit disk in $\mathbb {C}$, then "semistable" means that the special fiber is a divisor with normal crossings. The semistable reduction theorem for curves was first proved by Deligne and Mumford; the proof relied on the semistable reduction theorem for abelian varieties.

References

• P. Deligne and D. Mumford. The irreducibility of the space of curves of given genus. Publ. Math. IHES, 36:75–109, 1969.
• 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