Semistable reduction theorem

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

^ Morrison 1984, § 1.
^ Grothendieck (1972), Théorème 3.6, p. 351
^ Deligne & Mumford 1969, Corollary 2.7.

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.

References

Further reading