Abstract:
This talk addresses the classification of finite abelian subgroups in the
automorphism groups
of rationally connected varieties. This is a classical problem, with its
origins going back to
the late 19th century. An interesting dichotomy arises in dimension two: the
finite abelian
subgroups of Cremona group of rank 2 can be divided into two types. The first
type consists
of groups that can act on a Mori fiber space with non-trivial base (that is,
on a conic bundle).
The second, “exceptional” type, corresponds to elliptic curves with complex
multiplication
anti-canonically embedded in certain del Pezzo surfaces. We try to extend this
observation
to higher dimensions. In general, such exceptional abelian groups should
originate from
highly symmetric Calabi-Yau subvarieties found in birational modifications of
the original
rationally connected variety. In dimension 3, this role is played by anti-
canonically embedded
K3 surfaces of higher Picard rank, leading to a complete classification with
exactly four
exceptional groups. While these groups are realizable, their embedding into
the Cremona
group of rank 3 remains an open problem. We will also explore the extension
problem for
finite abelian groups and its connection to the geometry of 4-dimensional Mori
fiber spaces.
See also
|
