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Explore a mathematical lecture examining the classification of linearisable subgroups within the Cremona group for n=2. Delve into one of equivariant birational geometry's core challenges: classifying finite subgroups of birational automorphisms in n-dimensional projective space. Learn how the problem begins with identifying subgroups conjugate to those acting biregularly on n-dimensional projective space (subgroups of PGL(n+1)), known as linearisable subgroups. Discover why, although the Cremona group equals PGL(2) when n=1, the classification becomes significantly more complex for n=2, culminating in a comprehensive analysis of linearisable subgroups in this specific case.
