Flattening and algebrisation
Michael McQuillan (University of Rome Tor Vergata)
Friday 6th December, 2024 13:00-14:00 Maths 311B
Abstract
Often natural moduli problems, e.g. foliated surfaces, come without an ample line bundle, so the
algebraisability of formal deformations, and the very existence of a moduli space requires a study
of the mermorphic functions on the aforesaid deformations, and the flattening (by blowing up) of the
resulting meromorphic maps. In such a context the flattening theorem of Raynaud & Gruson, and derivatives thereof, is close to irrelevant since it systematically uses schemeness to globally
extend local centres of blowing up. This was already well understood by Hironaka in his proof of
holomorphic flattening, and his ideas are the right ones. Nevertheless, the said ideas can be better
organised with a more systematic use of Grothendieck's universal flatifier, and, doing so, leads to
a fully functorial, and radically simpler, proof provided the sheaf of nilpotent functions is
coherent-which is true for excellent formal schemes, but, unlike schemes or complex spaces, is false
in general.
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