Thomas Ducat

The decomposition groups of plane conic and rational cubic curves

The decomposition group of a plane curve C is the subgroup
of elements of the Cremona group which restrict to a birational map of
C. Hedén and Zimmermann proved that when C is a line then this group
is generated by linear maps and one elementary quadratic map
preserving C - an analogy of the Noether-Castelnuovo theorem for the
full Cremona group. Using this result we show that the decomposition
group is also generated by linear and quadratic maps when C is a conic
or rational cubic curve, but not in general for plane rational curves
of degree ≥4. This is joint work with Isac Hedén and Susanna
Zimmermann.