QUOTIENTS OF CONIC BUNDLES

Abstract

Let k be an arbitrary field of characteristic zero. In this paper we study quotients of k-rational conic bundles over by finite groups of automorphisms. We construct smooth minimal models for such quotients. We show that any quotient is birationally equivalent to a quotient of other k-rational conic bundle cyclic group $$ {\mathrm{\mathfrak{C}}}_{2^k} $$ of order 2 k , dihedral group $$ {\mathfrak{D}}_{2^k} $$ of order 2 k , alternating group $$ {\mathfrak{A}}_4 $$ of degree 4, symmetric group $$ {\mathfrak{S}}_4 $$ of degree 4 or alternating group $$ {\mathfrak{A}}_5 $$ of degree 5 effectively acting on the base of the conic bundle. Also we construct infinitely many examples of such quotients which are not k-birationally equivalent to each other.