## 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)].

### MSC:

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

### Citations:

Zbl 0573.13013; Zbl 0617.14034
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### References:

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