Nagata’s compactification and factorization of morphisms (Ilya Tyomkin, Ben-Gurion University of the Negev)

09.05.2016 17:30 – 18:15

After constructing first examples of complete non-projective varieties in late 50s, Nagata became interested in the compactification problem in algebraic geometry. In early 60s, he managed to prove that any variety, or more generally, a separated scheme of finite type over a Noetherian base scheme can be embedded as an open subscheme in a complete scheme over the base. This fundamental result is known as Nagata’s compactification theorem.

Reading Nagata’s original papers is not an easy task, since they were written in his own version of pre-Grothendieck language. In 70s, Deligne prepared notes written in modern terms providing a version of Nagata’s approach. He sent them to Nagata, but didn’t make them public for almost 40 years till 2010.

In 2009, Temkin found a new proof of Nagata’s compactification. He proved the following remarkable decomposition theorem: any separated morphism of qcqs schemes can be presented as a composition of an affine morphism followed by a proper morphism. Temkin deduced Nagata’s compactification from the decomposition theorem, since affine morphisms of finite type are easily compactifiable.

Today, several proofs of various versions of Nagata’s theorem are known: In addition to the proofs mentioned above, there is a short proof due to Lütkebohmert (1993) for the Noetherian base, there is a detailed version of Deligne’s proof without the Noetherianity assumption due to Conrad (2007), there are two recent proofs in the category of algebraic spaces due to Conrad, Lieblich, Olsson, and to Temkin and myself. Finally, there is an unfinished work of Rydh in the case of (some) Deligne-Mumford stacks.
In my talk, I’ll explain the problem, its history, and development. I’ll also try to give some hints about existing approaches and main difficulties.

Lieu

Bâtiment: Battelle

Villa Battelle, séminaire Battelle

entrée libre