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The decomposition, inertia and ramification groups in birational geometry. (English) Zbl 0834.14006
Tikhomirov, Alexander (ed.) et al., Algebraic geometry and its applications. Proceedings of the 8th algebraic geometry conference, Yaroslavl’, Russia, August 10-14, 1992. Braunschweig: Vieweg. Aspects Math. E 25, 39-45 (1994).
Let \(X\) be an irreducible scheme, \(G\) be a subgroup of the group \(\text{Bir} (X)\) of birational transformations of \(X\), \(Y\) be an irreducible reduced subscheme of \(X\), \(G_Y\) be the stabilizer in \(G\) of the generic point of \(Y\) (another name for \(G_Y\) is the decomposition group of \(Y\) in \(G)\). – The \(i\)-th ramification group of \(Y\) in \(G\) consists of those elements of \(G_Y\) that operate trivially on the \(i\)- th infinitesimal neighbourhood of \(Y\) in \(X\). Here, \(i\) is a nonnegative integer. The group \(G_{0Y}\) has another name, the inertia group. These notions generalize the classic ones from the theory of Dedekind domains as well as the notion of congruence subgroup.
There are three main subjects in this short paper.
The first one is a reduction of the problem of the simplicity of the plane Cremona group to an analogue of the congruence subgroup problem (certainly, both problems are difficult and unsolved).
The second topic is a connection between the group \(\text{Bir} (Y)\) of birational transformations of a minimal cubic surface \(Y \subset \mathbb{P}_3\) over a perfect field and the decomposition group \((\text{Bir} (\mathbb{P}_3))_Y\) of \(Y\). Surjectivity of a natural restriction homomorphism of the last group into the former one is proven. Moreover, this epimorphism has a section homomorphism in the opposite direction.
The third subject is the non-triviality of some ramification groups. If \(Y\) is an irreducible hypersurface of degree \(d\) defined over a field \(k\), \(Y \subset \mathbb{P}_n\), and the set of points \(P \in \mathbb{P}_n (k)\) with \(\text{mult}_P(Y) = d - 2\) is infinite (for example \(Y\) is a cubic hypersurface with infinite set of rational points), then all the groups \(\text{Bir} (\mathbb{P}_n/k)_{iY}\) are non-trivial.
For the entire collection see [Zbl 0793.00016].

14E07 Birational automorphisms, Cremona group and generalizations
14E05 Rational and birational maps