Analytic compactifications of C² part II - one irreducible curve at infinity

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. In particular we show that in 'most' of these surfaces C^2 is 'rigidly embedded'. As an application we give a description of 'embedded isomorphism classes' of planar curves with one place at infinity. We also compute the canonical divisor of these surfaces; it turns out that their log discrepancy is related to the Frobenius number of the semigroup of poles along the curve at infinity. We use the computation to classify Gorenstein primitive compactifications of C^2 with rational and minimally elliptic singularities, extending a result of Brenton, Drucker and Prins (Ann. of Math. Stud., vol 100, 1981). As another application we characterize weighted projective spaces of the form P^2(1,1,q) in terms of their 'log discrepancy' and 'index', generalizing a characterization of P^2 due to Borisov (Journal of Algebraic Combinatorics, 2014).