Normal hypersurfaces as a compactification of \(\mathbb{C}^2\) (Q2731417)

Let \(X_{d}\) be a normal Gorenstein hypersurface of degree \(d\) in \(\mathbb P^{3}(\mathbb C)\) and \(Y_{d}\) an irreducible closed subvariety of \(X_{d}\) such that \(X_{d}\backslash Y_{d}\) is biholomorphic to \(\mathbb C^{2}\). NEWLINENEWLINENEWLINEM. Furushima analysed the structure of \(X_{d}\) and the minimal resolution of \(X_{d}\) for \(2\leq d\leq 4\) [Math. Nachr. 186, 115-129 (1997; Zbl 0882.32010)], and in the paper under review the author extends these results, proving that the following is true for any \(d\geq 3\): \(Y_{d}\) is a line in \(\mathbb P^{3}\), \(X_{d}\) has at most two singular points and one of them has geometric genus \(p_{g}(x)=(d-1)(d-2)(d-3)/6\). Moreover the author describes in detail the graph of a minimal resolution of the singularities of \(X_{d}\), which is a tree of smooth rational curves. If \(X_{d}\) contains a point of multiplicity \(d-1\), then \(X_{d}\) has exactly one isolated singularity.