Irregularity of quintic surfaces of general type (Q1335284)

In this paper we prove the following theorem. Let \(X\) be a normal quintic surface in \(\mathbb{P}^3\) and let \(\widetilde X\) be a resolution of \(X\). If \(\widetilde X\) is of general type, then its irregularity vanishes. This result was proved also by \textit{J.-G. Yang} [Trans. Am. Math. Soc. 295, 431-473 (1986; Zbl 0596.14029)] as a corollary of a classification of singularities on \(X\). To the contrary our proof is direct. We study invariants of \(\widetilde X\) and then find a contradiction if the irregularity is not zero. Our method can be also applied to other surfaces of lower degree. In fact, we apply this method to prove the nonexistence of normal quintic surfaces which are birationally equivalent to an abelian or a hyperelliptic surface [\textit{I. Nakamura} and \textit{Y. Umezu}, ``Nonexistence of normal quintic abelian surfaces in \(\mathbb{P}^3\)'', Tokyo J. Math. 18, No. 2, 369-382 (1995)].