On the inertia group of elliptic curves in the Cremona group of the plane

Details On the inertia group of elliptic curves in the Cremona group of the plane On the inertia group of elliptic curves in the Cremona group of the plane

2007-03-27

arxiv.org/abs/math/0703804v2

Michigan Math. J. 56 (2008), no. 2, 315-330.

Mathematics

Algebraic Geometry

14 pages, no figure

Scientific paper

10.1307/mmj/1224783516

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 if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. 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.

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