Abstract
An Ulrich sheaf on an n-dimensional projective variety \(X \subseteq \mathbb {P}^{N}\) is an initialized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby–Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves \(\delta \)-Ulrich. In the case \(n=2,\) where \(\delta \)-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection to \(\mathbb {P}^2\) is a semistable instanton bundle on \(\mathbb {P}^{2}\), we show that some high Veronese embedding of X admits a \(\delta \)-Ulrich sheaf with a global section.
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References
Aprodu, M., Costa, L., Miro-Roig, R.: Ulrich bundles on ruled surfaces. Preprint arXiv:1609.08340
Aprodu, M., Farkas, G., Ortega, A.: Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces. J. Reine Angew. Math. doi:10.1515/crelle-2014-0124
Beauville, A.: Ulrich bundles on abelian surfaces. Proc. Am. Math. Soc. 144(11), 4609–4611 (2016)
Backelin, J., Herzog, J., Sanders, H.: Matrix factorizations of homogeneous polynomials. In: Avramov, L., Tchakerian, K. (eds.) Algebra Some Current Trends (Varna, 1986), Volume 1352 of Lecture Notes in Mathematics, pp. 1–33. Springer, Berlin (1988)
Brennan, J.P., Herzog, J., Ulrich, B.: Maximally generated Cohen–Macaulay modules. Math. Scand. 61(2), 181–203 (1987)
Backelin, J., Herzog, J., Ulrich, B.: Linear maximal Cohen–Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71(2–3), 187–202 (1991)
Borisov, L., Nuer, H.: Ulrich bundles on Enriques surfaces. Preprint arXiv:1606.01459
Childs, L.N.: Linearizing of \(n\)-ic forms and generalized Clifford algebras. Linear Multilinear Algebra 5(4), 267–278 (1977/1978)
Coskun, I., Huizenga, J., Woolf, M.: Equivariant Ulrich Bundles on Flag Varieties. Preprint arXiv:1507.00102
Costa, L., Miró-Roig, R.M.: \(GL(V)\)-invariant Ulrich bundles on Grassmannians. Math. Ann. 361(1–2), 443–457 (2015)
Costa, L., Miró-Roig, R.M., Pons-Llopis, J.: The representation type of Segre varieties. Adv. Math. 230(4–6), 1995–2013 (2012)
Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16(3), 537–579 (2003)
Faenzi, D., Pons-Llopis, J.: The CM representation type of projective varieties. Preprint arXiv:1504.03819
Hanes, D.: Special Conditions on Maximal Cohen–Macaulay Modules, and Applications to the Theory of Multiplicities. University of Michigan, Ph.D. thesis (1999)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (2010)
Jardim, M.: Instanton sheaves on complex projective spaces. Collect. Math. 57(1), 69–91 (2006)
Kulkarni, R.S., Mustopa, Y., Shipman, I.: The characteristic polynomial of an algebra and representations. Preprint arXiv:1507.08361
Kulkarni, R.S., Mustopa, Y., Shipman, I.: Ulrich sheaves and higher-rank Brill–Noether theory. J. Algebra 474, 166–179 (2017)
Migliore, J., Nagel, U.: Survey article: a tour of the weak and strong Lefschetz properties. J. Commut. Algebra 5(3), 329–358 (2013)
Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62(1), 153–164 (2013)
Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel (2011). Corrected reprint of the 1988 edition with an appendix by S. I. Gelfand
Pappacena, C.J.: Matrix pencils and a generalized Clifford algebra. Linear Algebra Appl. 313(1–3), 1–20 (2000)
Procesi, C.: A formal inverse to the Cayley–Hamilton theorem. J. Algebra 107(1), 63–74 (1987)
Roby, N.: Algèbres de Clifford des formes polynomes. C. R. Acad. Sci. Paris Sér A–B 268, 484–486 (1969)
Van den Bergh, M.: Linearisations of binary and ternary forms. J. Algebra 109(1), 172–183 (1987)
Acknowledgements
I.S. was partially supported during the preparation of this paper by National Science Foundation award DMS-1204733. R. K. was partially supported by the National Science Foundation awards DMS-1004306 and DMS-1305377. We would like to thank the referee for helpful comments.
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Kulkarni, R.S., Mustopa, Y. & Shipman, I. Vector bundles whose restriction to a linear section is Ulrich. Math. Z. 287, 1307–1326 (2017). https://doi.org/10.1007/s00209-017-1869-0
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DOI: https://doi.org/10.1007/s00209-017-1869-0