Nonexistence of normal quintic abelian surfaces in \(\mathbb{P}^ 3\)
Normal surfaces \(X_d\) of degree \(d\) in the complex projective 3-space \(\mathbb{P}^3\) have simple birational structure if \(d\) is small: \(X_1\) and \(X_2\) are rational, and \(X_3\) is birationally equivalent to a ruled surface, since in general \(K_{X_d} \simeq {\mathcal O}_{X_d} (d-4)\). Moreover \(X_4\) is birationally equivalent to either a ruled or a K3 surface. To the contrary, various \(X_d\) may occur if \(d\geq 5\). If the singularity of \(X_d\) is mild, then \(X_d\) is birationally equivalent to a surface of general type, while \(X_d\) may be birationally equivalent to a ruled surface if it has severe singularity. Moreover there are examples of \(X_5\) which are birationally equivalent to K3 surfaces, Enriques surfaces or general elliptic surfaces. This leads us to the question whether there exists an \(X_d\) which is birationally an abelian or a hyperelliptic surface or not. The purpose of this note is to answer this question in the case of \(d=5\). We prove: Main theorem. No normal quintic surface in \(\mathbb{P}^3\) is birationally equivalent to an abelian or a hyperelliptic surface. Our proof of the theorem goes as follows. First we note that if a normal quintic surface \(X=X_5\) is birationally an abelian or a hyperelliptic surface, then its minimal resolution \(\widetilde X\) is an at most 5-fold blowing-up \(\mu: \widetilde X\to \overline X\) of the non-singular minimal model \(\overline X\). On the other hand, the pull-back of \(K_X\) to \(\widetilde X\) minus \(K_{\widetilde X}\) is an effective divisor \(\widetilde D\), which reflects the property of the singularity of \(X\) fairly well. Such property of \(\widetilde D\) and the condition of \(\mu_* \widetilde D\) as a divisor on an abelian or hyperelliptic surface finally lead us in every case to a contradiction. Conjecture. No normal hypersurface in \(\mathbb{P}^3\) is birationally equivalent to an abelian surface. Problem. Are there normal hypersurfaces in \(\mathbb{P}^3\) which are birationally hyperelliptic surfaces?
scientific article; zbMATH DE number 880003