Polynomial Automorphisms and Birational Transformations – BirPol

The activities planned in this project (quarterly
mini-workshops, thematic conferences, and an international congress) have two complementary objectives: first to create an environment which encourages new collaborations between the various domains of the experts of the team which ould lead to new results, and secondly to develop and promote the following directions of research:

1) the use and daptation of the tools of Mori theory to the studies of polynomial and birational automorphisms; and
2) the evelopment of a new approach to the moduli space of infinite dimensional groups based on the existing theory of ind-varieties and algebraic ind-schemes.

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This project also aims to contribute to the establishment of a network of French researchers around the themes of affine geometry and birational transformations. At this time, such a network exists, but it
is only structured around an informal Inter-University Franco-Swiss workshop group created in 2006 on the theme of
automorphisms of affine spaces. This project will develop the group's activities, promote and distribute its interests and
increase its visibility both nationally and internationally.

Transform. Groups 17 (2012), no. 1, 21-50.
[2] J. Blanc et S. Lamy, Weak Fano threefolds obtained by blowing-up a space curve and
construction of Sarkisov links, Proc. Lond. Math. Soc. (à paraître).
[3] S. Cantat & I. Dolgachev, Rational Surfaces with a Large Group of Automorphisms; J.
Amer. Math. Soc., vol 25 (2012), no. 3, 863-905
[4] A. Dubouloz & S. Lamy, Automorphisms of open surfaces with irreducible boundary,
preprint Février 2012.
[5] E. Edo, T. Kanehira, M. Karas & S. Kuroda, Separability of wild automorphisms of a
polynomial ring, preprint 2012.
[6] S. Lamy & J. Sebag, Birational self-maps and piecewise linear algebraic geometry, preprint
decembre 2011.
[7] S. Lamy & S. Vénéreau, The tame and the wild automorphisms of an affine quadric
threefold, Journal of the Mathematical Society of Japan (à paraître).

The project BirPol concerns the interaction of algebraic geometry, commutative algebra, holomorphic dynamics and geometric group theory. The aim is to study the groups of polynomial automorphisms of affine spaces and Cremona groups, that is, birational transformations of projective spaces. These groups have already been extensively studied since the 19th century in classical algebraic geometry, however there are many questions concerning their structure which are still open for dimension three and higher. The study of these two groups has long been conducted separately, guided by varying interests and approaches. However, many recent results, both in algebraic geometry, group theory and holomorphic dynamics reveal a profound analogy between the two subjects both at the levels of known results essentially in dimension two, as in the conceptual and technical difficulties encountered in addressing their study in higher dimension.

The main scientific objective is to develop common techniques and approaches allowing us on the one hand to reinterpret and improve the existing results in dimension 2 and secondly to make significant progress in the study of polynomial automorphisms and birational automorphisms in dimension 3. The project team consists therefore of experts from various complementary aspects of these questions (birational projective geometry, affine algebraic geometry, holomorphic dynamics, real algebraic geometry, commutative algebra, invariant theory, geometric group theory), who will collaborate on different aspects to develop new approaches in this research area.

The activities planned in this project (quarterly mini-workshops, thematic conferences, and an international congress) have two complementary objectives: first to create an environment which encourages new collaborations between the various domains of the experts of the team which could lead to new results, and secondly to develop and promote the following directions of research :
1) the use and adaptation of the tools of Mori theory to the studies of polynomial and birational automorphisms; and
2) the development of a new approach to the moduli space of infinite dimensional groups based on the existing theory of ind-varieties and algebraic ind-schemes.

This project also aims to contribute to the establishment of a network of French researchers around the themes of affine geometry and birational transformations. At this time, such a network exists, but it is only structured around an informal Inter-University Franco-Swiss workshop group created in 2006 on the theme of automorphisms of affine spaces. This project will develop the group's activities, promote and distribute its interests and increase its visibility both nationally and internationally.