Birational classification of minimal del Pezzo surfaces

Surfaces with a conic bundle structure with Picard number 2 and del Pezzo surfaces with Picard number 1 naturally appear as a result of minimal model program for geometrically rational surfaces over arbitrary field. There is a natural question: which minimal surfaces are birational equivalent to a given minimal geometrically rational surface. One can apply theory of Sarkisov links to answer this question: any birational map between minimal surfaces can be decomposed into a sequence of Sarkisov links. These links are completely described, therefore for any minimal del Pezzo surface one can describe minimal surfaces birationally equivalent to this surface. For example, this method allows to prove Iskoskikh rationality criterion, that describes which minimal surfaces are rational. But the description of Sarkisov links allows to describe only degree of obtained surfaces but not allow to describe such surfaces up to isomorphism. In the talk we discuss some recent results which allow to describe all minimal surfaces birationally equivalent to a given minimal del Pezzo surfaces up to isomorphism. In particularly, we show that for any given pointless del Pezzo surface of degree 8 or del Pezzo surface of degree 4 with invariant Picard number 1 any del Pezzo surface with Picard number 1 birationally equivalent to the given one is isomorphic to the given one.