Semistable reduction theorem explained
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.
[1] 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 important 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
- Deligne . P. . Mumford . D. . The irreducibility of the space of curves of given genus . Publications Mathématiques de l'Institut des Hautes Scientifiques . 36 . 75–109 . 1969 . 36 . 10.1007/BF02684599. 16482150 .
- Book: Grothendieck
, Alexandre
. Groupes de Monodromie en Géométrie Algébrique . Alexandre Grothendieck . Lecture Notes in Mathematics . 288 . 1972 . . Berlin; New York . fr . viii+523 . true . 10.1007/BFb0068688 . 978-3-540-05987-5 . 0354656.
- Book: http://web.math.ucsb.edu/~drm/papers/clemens-schmid.pdf. 125739605 . 10.1515/9781400881659-007 . Chapter VI. The Clemens-Schmid exact sequence and applications . Topics in Transcendental Algebraic Geometry. (AM-106) . 1984 . Morrison . David R. . 101–120 . 9781400881659 .
Further reading
Notes and References
- Grothendieck (1972), Théorème 3.6, p. 351