Rational curves on quasi-projective surfaces

Introduction and statement of results Glossary of notation and conventions Gorenstein del Pezzo surfaces Bug-eyed covers Log deformation theory Criteria for log uniruledness Reduction to πᵃˡᵍ₁(S⁰)=\1\ Flushness and preparation for the hunt Bogomolov bound Riemann Roch and surfaces with small coefficient A partial classification of KT-contractions the linear system KS + A Classification of bananas and fences T₁ a net g(A₁)>1 A₁ has a simple cusp A₁ has a simple node A₁ smooth The smooth banana Proof of (1.1) and corollaries A surface with πᵃˡᵍ₁(S⁰)=\1\ but no tiger Tigers, complements and toric pairs Classification of almost all rank one log del Pezzos Appendix: Log terminal surface singularities and adjunction Appendix: Normalisation of an algebraic space Index to obscure or unconventional notation References.