Normal affine surfaces with \(\mathbb{C}^*\)-actions (Q1423474)

Given a normal affine surface \(X\) with a nontrivial \(\mathbb{C}^{*}\)-action, the coordinate algebra \(A=\mathbb{C}[X]\) comes equipped with a \(\mathbb{Z}\)-grading. There are three basic types of gradings and \(\mathbb{C}^{*}\)-actions: \(A_0=\mathbb{C}\) and \(A_{+}\) or \(A_{-}=0\) (elliptic case); \(A_0\neq\mathbb{C}\) and \(A_{+}\) or \(A_{-}=0\) (parabolic case); \(A_{\pm}\neq0\) (hyperbolic case). \textit{H. Pinkham} [Math. Ann. 227, 183--193 (1977; Zbl 0338.14010)] and \textit{M. Demazure} [Trav. Cours 37, 35--68 (1988; Zbl 0686.14005)] proved that in elliptic and parabolic case \(A=\bigoplus_{n\geq0}H^0(C,[nD])\) for a certain projective or affine curve \(C\) and a \(\mathbb{Q}\)-divisor \(D\) on \(C\), and the graded isomorphism class of \(A\) is determined by the rational equivalence class of \(D\). In the parabolic case, the Pinkham--Demazure construction is used in the paper to show that each rational parabolic \(\mathbb{C}^{*}\)-surface is realized as the normalization of a surface in \(\mathbb{A}^3\) given by an equation \(x^d=P(z)y\). Also, the Pinkham--Demazure construction is generalized in the paper to hyperbolic case in the following form: coordinate algebras of hyperbolic \(\mathbb{C}^{*}\)-surfaces are of the form \(A=\bigoplus_{n\geq0}H^0(C,[nD_{+}])\oplus\bigoplus_{n<0}H^0(C,[nD_{-}])\), where \(D_{\pm}\) are \(\mathbb{Q}\)-divisors on an affine curve \(C\) determined up to a simultaneous shift by a principal divisor and such that \(D_{+}-D_{-}\leq0\). This is used to realize rational hyperbolic \(\mathbb{C}^{*}\)-surfaces as normalizations of surfaces in \(\mathbb{A}^4\) given by equations of certain type, to determine the structure of singularities, the \(\mathbb{C}^{*}\)-orbits, the divisor class group, and the canonical divisor of a hyperbolic surface.