Logarithmic plurigenera of smooth affine surfaces with finite Picard groups
The author proves some interesting results about dimensions of logarithmic pluricanonical systems. Let \(S\) be a smooth affine surface with finite Picard group. Let \(\overline P_n(S)\) denote \(\dim H^0(\overline S, n(K+ D)\), where \(\overline S\) is a smooth projective completion of \(S\) such that \(D= \overline S- S\) has simple normal crossings. Let \(\overline k(S)\) be the logarithmic Kodaira dimension of \(S\) as defined by S. Iitaka. The following results are proved. 1. If \(\overline k(S)= 1\) then \(\overline P_2(S)> 0\). If \(\overline k(S)= 2\) then \(\overline P_6(S)> 0\). 2. Structure of \(S\) is determined when \(\overline k(S)\geq 0\) and \(\overline P_6(S)= 0\). 3. If \(\text{Pic}(S)= (0)\), \(\Gamma(S,{\mathcal O}_S)^*= \mathbb{C}^*\) and \(\overline P_2(S)= 0\) then \(S\approx\mathbb{C}^2\). Earlier results analogous to (1), (2), were proved by Kuramoto and Tsunoda. (3) is related to the proof of the cancellation theorem for \(\mathbb{C}^2\) by Miyanishi-Sugie-Fujita. The proofs use the theory of Open Algebraic Surfaces developed by Japanese mathematicians.
affine surfaces with finite Picard groups
scientific article; zbMATH DE number 5311214