Remarks on higher-rank ACM bundles on hypersurfaces
This paper deals with the existence of arithmetically Cohen-Macaulay (aCM) bundles \(\mathcal{E}\) on smooth hypersurfaces \(X_d\subset \mathbb{P}^{n+1}\) of degree \(d\). It was conjectured by Buchweitz, Greuel and Schreyer [\textit{R. O. Buchweitz} et al., Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)] (motivated by the classification of aCM bundles on hyperquadrics) that if \(r:=\mathrm{rank}(\mathcal{E})<2^e\), where \(e:=\lfloor\frac{n-1}{2}\rfloor\), then \(\mathcal{E}\) splits as a direct sum of line bundles. Based on some splitting results, the first named author proposed a modified version of the aforementioned conjecture: for a general hypersurface \(X_d\subset \mathbb{P}^{n+1}\) of sufficiently high degree, an aCM bundle of rank \(r\) should split if \(r< 2^s\), where \(s:=\lfloor\frac{n+1}{2}\rfloor\). In this paper, the authors continue contributing to this conjecture proving that on a general hypersurface \(X_d\subset \mathbb{P}^5\) of degree \(d\geq 3\) there are no indecomposable rank-\(3\) aCM bundles with fewer than \(8\) generators (Theorem 1 and Corollary 1). Moving to the opposite case of aCM bundles with the maximum permitted number \(r.d\) of generators (i.e, the so-called Ulrich bundles), the authors prove that a smooth hypersurface of even degree does not support an Ulrich bundle of odd rank and determinant of the form \(\mathcal{O}_X(c)\) for \(c\in\mathbb{Z}\) (Theorem 2).
scientific article; zbMATH DE number 6994063
scientific article