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Remarks on higher-rank ACM bundles on hypersurfaces. (Remarques sur les fibrés ACM de rang supérieur sur les hypersurfaces.) (English. French summary) Zbl 1403.14078

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 [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).

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
13C14 Cohen-Macaulay modules
14J70 Hypersurfaces and algebraic geometry

Citations:

Zbl 0617.14034
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References:

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