##### Department of Mathematics,

University of California San Diego

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### Math 208: Seminar in Algebraic Geometry

## Dr. Michael McQuillan

#### University of Rome Tor Vergata

## Flattening and algebrisation.

##### 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.

Host: James McKernan

### May 31, 2024

### 4:00 PM

APM 7321

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