The essential and the Cremona dimensions of a group

2025/07/20 by Dolgachev, Igor
#14E07 #14L30 #20G10 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.15096

Abstract

The Cremona dimension of a group G is the minimal n such that G is isomorphic to a subgroup of the Cremona group of birational transformations of an n-dimensional rational variety. In this survey article, we give many examples that gives evidence to the conjecture that the Cremona dimension of a finite group over the field of complex numbers is less than or equal to the essential dimension of the group.

Citations

• Essential dimension of isogenies
• Essential dimension of symmetric groups in prime characteristic
• Automorphisms of del Pezzo surfaces in odd characteristic
• Embeddings of the symmetric groups to the space Cremona group
• The essential p-dimension of the split finite quasi-simple groups of classical Lie type
• Finite quasisimple groups acting on rationally connected threefolds
• Finite quasisimple groups acting on rationally connected threefolds
• Essential dimension in mixed characteristic
• Quartic surfaces with icosahedral symmetry
• Versality of algebraic group actions and rational points on twisted\n varieties
• Non-rationality of the symmetric sextic Fano threefold
• Finite simple groups of small essential dimension
• On the essential dimension of infinitesimal group schemes
• The Coble hypersurfaces
• Essential dimensions of algebraic groups and a resolution theorem for G-varieties
• On the projective geometry of homogeneous spaces
• Elementary abelian p-subgroups of algebraic groups

Related

• The Jordan property of Cremona groups and essential dimension
• Representations of finite subgroups of Cremona groups
• Categorical dimension of birational automorphisms and filtrations of Cremona groups
• On homomorphisms between Cremona groups
• On abelian extensions of finite abelian subgroups of Cremona groups