Analytic compactifications of C^2 II: one irreducible curve at infinity

Abstract

We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic surfaces containing C^2 for which the curve at infinity is irreducible), compute the moduli space of these surfaces and their groups of auomorphisms. As a result we show that in 'most' of these surfaces C^2 is 'rigidly embedded'. We also give a description of 'embedded isomorphism classes' of planar curves with one place at infinity. We compute the canonical divisor and find numerical characterizations of primitive compactifications with at most rational and elliptic singularities, and those which are Gorenstein. This in particular recovers the classification by Brenton, Drucker, and Prins ("Graph theoretic techniques in algebraic geometry", volume 100, Ann. of Math. Stud., 1981) of Gorenstein primitive compactifications of C^2 with vanishing geometric genus. As an application of our results we classify all G^2_a-varieties with Picard number 1 - which completes the first step towards answering a question of Hassett and Tschinkel ("Geometry of equivariant compactifications of G_a^n", Internat. Math. Res. Notices, 1999).