Abstract
The objective is to show the construction of an Ulrich vector bundle on the blowing-up \({\widetilde{X}}\) of a nonsingular projective variety X at a closed point, where the original variety is embedded by a very ample divisor H and carries an Ulrich vector bundle. In order to achieve this result, we aim to find a suitable very ample divisor on \({\widetilde{X}}\), which is dependent on H. At the end, we take into consideration some applications to surfaces with regards to minimal models and their Kodaira dimension.
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References
Aprodu, M., Costa, L., Miró-Roig, R.M.: Ulrich bundles on ruled surfaces. J. Pure Appl. Algebra 222(1), 131–138 (2018)
Beauville, A.: Complex Algebraic Surfaces. London Mathematical Society Student Texts, vol. 34. Cambridge University Press, Cambridge (1996)
Beauville, A.: Determinantal hypersurfaces. Mich. Math. J. 48, 39–64 (2000)
Beauville, A.: Ulrich bundles on Abelian surfaces. Proc. Am. Math. Soc. 144(11), 4609–4611 (2016)
Beauville, A.: Ulrich bundles on surfaces with \(p_g=q=0\). Preprint arXiv:1607.00895 (2016)
Beauville, A.: An introduction to Ulrich bundles. Eur. J. Math. 4(1), 26–36 (2018)
Beltrametti, M.C., Sommese, A.J.: Notes on embeddings of blowups. J. Algebra 186, 861–871 (1996)
Bertram, A., Ein, L., Lazarsfeld, R.: Vanishing theorems, a theorem of severi, and the equation defining projective varieties. J. Am. Math. Soc. 4(3), 587–602 (1991)
Casnati, G.: Special Ulrich bundles on non-special surfaces with \(p_g=q=0\). Int. J. Math. 28(8), 1750061 (2017)
Casnati, G.: Ulrich bundles on non-special surfaces with \(p_g=0\) and \(q=1\). Revista Matemática Complutense (2017). https://doi.org/10.1007/s13163-017-0248-z
Eisenbud, D., Schreyer, F.-O.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16(3), 537–579 (2003)
Faenzi, D.: Ulrich Bundles on K3 Surfaces. Preprint arXiv:1807.07826 (2018)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Herzog, J., Ulrich, B., Backelin, J.: Linear maximal Cohen-Macaulay modules over strict complete intersections. J. Pure Appl. Algebra 71(2–3), 187–202 (1991)
Kim, Y.: Ulrich bundles on blowing ups. C. R. Math. Acad. Sci. Paris 354(12), 1215–1218 (2016)
Lopez, A. F.: On the existence of Ulrich vector bundles on some irregular surfaces. Preprint arXiv:1812.10195v4 (2018)
Ulrich, B.: Gorenstein rings and modules with high numbers of generators. Math. Z. 188(1), 23–32 (1984)
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Secci, S.A. On the existence of Ulrich bundles on blown-up varieties at a point. Boll Unione Mat Ital 13, 131–135 (2020). https://doi.org/10.1007/s40574-019-00208-6
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DOI: https://doi.org/10.1007/s40574-019-00208-6