2025/01/30 by Abouelsaad, Ahmed
#14E05 #14E07 #14H05 #14H50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2501.18551
Let \Cr_\Q(2) be the Cremona group of rank 2 over rational numbers. We give a classification of large finite subgroups G of \Cr_\Q(2) and give a new sharp bound smaller (but not multiplicative) than M(\Q)=120960 = 27⋅33⋅5⋅7; the one given in \citeMR2567402. In particular, we prove that any finite subgroup G ⊂\Cr_\Q(2) has order | G| ≤ 432 and Lemma \reflemm-25 provides a group of order 432. We use the modern approach of minimal G-surfaces, given a (smooth) rational surface S⊂\p2 defined over \Q, we study the finite subgroups G ⊂ \Aut\Q(S) of automorphisms of S. We give the best bound for the order of G⊂\Aut(S) for surfaces with a conic bundle structure invariant by G. We also give the best bound for the order of G⊂ \Aut_\Q(S) for all rational Del Pezzo surfaces of some given degree. In addition, we give descriptions of the finite subgroups of automorphisms of conic bundles and Del Pezzo surfaces of maximal size.