A note on smooth surfaces in \(\mathbb{P}^4\) (Q1392735)

\textit{G. Ellingsrud} and \textit{Ch. Peskine} [Invent. Math. 95, No, 1, 1-11 (1989; Zbl 0676.14009)] proved a theorem which states that there are only a finite number of components in the Hilbert scheme parameterizing smooth surfaces in \(\mathbb{P}^4\) not of general type. On the other hand, \textit{R. Braun} and \textit{G. Fløystad} [Compos. Math. 93, No. 2, 211-229 (1994; Zbl 0823.14021)] proved that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\). In this note, the author proves that there is only a finite number of components in the Hilbert scheme of surfaces in \(\mathbb{P}^r\) parameterizing integral surfaces of degree \(d\). Then he deduces that a smooth surface in \(\mathbb{P}^4\) not of general type has degree \(d\leq 105\). Thus he obtains a simpler and quicker proof of the above theorem. The proof in this note essentially relies on Castelnuovo's theory and on a lower bound for the genus of the generic hyperplane section of a smooth surface in \(\mathbb{P}^4\).