Resolution of indeterminacy of pairs. (English) Zbl 1098.14008

Beltrametti, Mauro C. (ed.) et al., Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter (ISBN 3-11-017180-5/hbk). 165-177 (2002).
If \(G\) is a finite group of birational isomorphisms of the smooth projective variety \(Y\), then a resolution of the indeterminacy of the pair \((Y,G)\) is any birational map \(\phi : X \rightarrow Y\) with \(X\) a smooth projective variety, such that for any \(\tau \in G\) the birational map \(\phi^{-1} \tau \phi : X \rightarrow X\) is biregular, i.e. an automorphism of \(X\). Such resolutions always exist (see section 1), and the problem is to study them more closely in some particular occurences.
In this paper is studied in some detail the case when \(\dim \;Y = 2\), i.e. when \(Y\) is a smooth projective surface. In section 2 is established a correspondence between birational morphisms of smooth surfaces and finite closed subsets of algebraic valuations. These results are used in section 3 to prove the existence of a minimal resolution of a pair \((Y,G)\) in the case when \(\dim Y = 2\).
In the last section 4 is introduced a birational invariant of each subgroup of prime order of the group \(\text{Bir}(Y)\) of birational isomorphisms of a projective surface \(Y\). This invariant is used to determine when two subgroups of prime order of \(\text{Bir}({\mathbb P}^2)\) are conjugate. As mentioned in the paper, a close study of resolutions of indeterminacy of pairs in the case \(\dim\;Y = 3\) has been undertaken by I. A. Cheltsov [Math. Notes 76, No. 2, 264–275 (2004; Zbl 1059.14019)].
For the entire collection see [Zbl 0996.00060].


14E05 Rational and birational maps
14E07 Birational automorphisms, Cremona group and generalizations
14E15 Global theory and resolution of singularities (algebro-geometric aspects)


Zbl 1059.14019