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The representation type of rational normal scrolls. (English) Zbl 1268.14014
Let \((X,\mathcal{O}_X(1))\) be a polarized projective variety of dimension \(n\). A sheaf \(E\) on \(X\) is called Arithmetically Cohen-Macaulay (ACM), if all its middle cohomologies vanish. That is, \(H^i(X,E(k))=0\) for all \(k\) and \(0<i<n\), where as usual, one writes \(E(k)\) to mean \(E\otimes\mathcal{O}_X^{\otimes k}\). For such an \(E\), one can see easily that the number of generators of \(\bigoplus H^0(X,E(k))\) is at most \(\deg X\cdot \mathrm{rk}\, E\). An \(E\) where equality holds are called Ulrich bundles and they have been studied intensely and introduced by B. Ulrich [Math. Z. 188, 23–32 (1984; Zbl 0573.13013)]. The paper under review shows that on most rational normal scrolls there are arbitrarily large rank and dimension families of indecomposable Ulrich bundles. The exceptions, a small number, were previously known to have not many indecomposable Ulrich bundles [R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165–182 (1987; Zbl 0617.14034)].

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14M20 Rational and unirational varieties
Full Text: DOI
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