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Finite Subgroups of the Plane Cremona Group
 IN ALGEBRA, ARITHMETIC AND GEOMETRY: MANIN FESTSCHRIFT (BIRKHÄUSER
, 2009
"... This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane. ..."
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Cited by 73 (6 self)
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This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.
Walks with small steps in the quarter plane
 Contemporary Mathematics
"... Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a mo ..."
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Cited by 47 (7 self)
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Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a halfplane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be Dfinite. The 23rd model, known as Gessel’s walks, has recently been proved by Bostan et al. to have an algebraic (and hence Dfinite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a nonDfinite generating function. Our approach allows us to recover and refine some known results, and also to obtain new
Convergence of the Kähler Ricci flow and multiplier ideal sheaves on Del Pezzo surfaces
"... Abstract. On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the KählerRicci flow with a certain initial value converges in C ∞norm exponentially fast to a KählerEinstein metric. The proof is based on the method of multiplier ideal sheaves. 1. ..."
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Cited by 7 (1 self)
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Abstract. On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the KählerRicci flow with a certain initial value converges in C ∞norm exponentially fast to a KählerEinstein metric. The proof is based on the method of multiplier ideal sheaves. 1.
ON THE INERTIA GROUP OF ELLIPTIC CURVES IN THE CREMONA GROUP OF THE PLANE
, 2007
"... Abstract. We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that the embedding of the fixed curve in P 2 need to be bir ..."
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Cited by 7 (2 self)
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Abstract. We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that the embedding of the fixed curve in P 2 need to be birationally equivalent to a plane smooth cubic. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve. 1.
Finite groups of essential dimension 2
 Comment. Math. Helv
"... We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0. Mathematics Subject Classification: 14E07, 14L30, 14J26 1 ..."
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Cited by 6 (4 self)
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We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0. Mathematics Subject Classification: 14E07, 14L30, 14J26 1
Some properties of the Cremona group
, 2012
"... We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the wellknown subgroup of the Cremona group made up of the polynomial automorphisms of C2. Then we deal with the classification of birational ma ..."
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Cited by 3 (0 self)
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We recall some properties, unfortunately not all, of the Cremona group. We first begin by presenting a nice proof of the amalgamated product structure of the wellknown subgroup of the Cremona group made up of the polynomial automorphisms of C2. Then we deal with the classification of birational maps and some applications (Tits alternative, nonsimplicity...) Since any birational map can be written as a composition of quadratic birational maps up to an automorphism of the complex projective plane, we spend time on these special maps. Some questions of group theory are evoked: the classification of the finite subgroups of the Cremona group and related problems, the description of the automorphisms of the Cremona group and the representations of some lattices in the Cremona group. The description of the centralizers of discrete dynamical systems is an important problem in real and complex dynamic, we describe the state of the art for this problem in the Cremona group. Let S be a compact complex surface which carries an automorphism f of positive topological entropy. Either the Kodaira dimension of S is zero and f is conjugate to an automorphism on the unique minimal model of S which is either a torus, or a K3 surface, or an Enriques surface, or S is a nonminimal rational surface and f is conjugate to a birational map of the complex projective plane. We deal with results obtained in this last case: construction of such automorphisms, dynamical properties (rotation domains...).
GEOMETRICALLY RATIONAL REAL CONIC BUNDLES AND VERY TRANSITIVE ACTIONS
, 2009
"... We study very transitive groups of automorphisms of real geometrically rational surfaces with applications to the classification of real algebraic models of compact surfaces. We give an insight into the geometry of real parts which is a geometry between biregular and birational geometry’s, and sho ..."
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Cited by 3 (2 self)
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We study very transitive groups of automorphisms of real geometrically rational surfaces with applications to the classification of real algebraic models of compact surfaces. We give an insight into the geometry of real parts which is a geometry between biregular and birational geometry’s, and show several surprising facts about it.