In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.

We compute the Chow group of 0-cycles on a rational surface defined over a finite extension K of the field $\mathbb{Q}_p$ of p-adic numbers (p a prime) when it is split by an unramified extension of K. We use intersection theory to define a specialisation map so we need to assume that the surface admits a regular proper integral model. A family of examples is worked out to illustrate the method.

Simply connected closed symplectic 4-manifolds with $b_{2}^+ \,{=}\,1$ and $K^2 \,{=}\,0$ are investigated. As a result, it is confirmed that most of homotopy elliptic surfaces $\{E(1)_{K} \,{|}\,K\, \mathrm{is\, a\, fibred\, knot}$ in $S^3\}$ constructed by R. Fintushel and R. Stern in Invent. Math. 134 (1998) 363–400 are simply connected closed minimal symplectic 4-manifolds that do not admit a complex structure.This work was supported by grant No. R14-2002-007-01002-0 from KOSEF.