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
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[1] Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of algebraic curves. Grundlehren der Mathematischen Wissenschaften, vol. 267. Springer, New York (1985) · Zbl 0559.14017
[2] Buchweitz, R., Greuel, G., Schreyer, F.O.: Cohen-Macaulay modules on hypersurface singularities, II. Invent. Math. 88(1), 165–182 (1987) · Zbl 0617.14034
[3] Casanellas, M., Hartshorne, R.: Stable Ulrich bundles, Preprint, available from arXiv: 1102.0878.
[4] Casanellas, M., Hartshorne, R.: Gorenstein Biliaison and ACM sheaves. J. Algebra 278, 314–341 (2004) · Zbl 1057.14062
[5] Casanellas, M., Hartshorne, R.: ACM bundles on cubic surfaces. J. Eur. Math. Soc. 13, 709–731 (2011) · Zbl 1245.14044
[6] Costa, L., Miró-Roig, R.M., Pons-Llopis, J.: The representation type of Segre varieties. Adv. Math. 230, 1995–2013 (2012) · Zbl 1256.14015
[7] Drozd, Y., Greuel, G.M.: Tame and wild projective curves and classification of vector bundles. J. Algebra 246, 1–54 (2001) · Zbl 1065.14041
[8] Eisenbud, D., Schreyer, F., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16, 537–579 (2003) · Zbl 1069.14019
[9] Eisenbud, D., Herzog, J.: The classification of homogeneous Cohen-Macaulay rings of finite representation type. Math. Ann. 280(2), 347–352 (1988) · Zbl 0616.13011
[10] Hartshorne, R.: Connectedness of the Hilbert scheme. Publications Mathmatiques de l’IHS 29, 5–48 (1966) · Zbl 0171.41502
[11] Horrocks, G.: Vector bundles on the punctual spectrum of a local ring. Proc. Lond. Math. Soc. 14(3), 689–713 (1964) · Zbl 0126.16801
[12] Miró-Roig, R.M.: On the representation type of a projective variety (preprint 2012) · Zbl 1327.14088
[13] Miró-Roig, R.M., Pons-Llopis, J.: N-dimensional Fano varieties of wild representation type (preprint) available from arXiv:1011.3704.
[14] Miró-Roig, R.M., Pons-Llopis, J.: Representation type of rational ACM surfaces $$X\(\backslash\)subseteq {\(\backslash\)mathbb{P}}\^4$$ (to appear in Algebras and Representation Theory) · Zbl 1277.14009
[15] Pons-Llopis, J., Tonini, F.: ACM bundles on Del Pezzo surfaces. Le Matematiche 64(2), 177–211 (2009) · Zbl 1207.14046
[16] Ulrich, B.: Gorenstein rings and modules with high number of generators. Math. Z. 188, 23–32 (1984) · Zbl 0573.13013
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