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RANK TWO STABLE ULRICH BUNDLES ON ANTICANONICALLY EMBEDDED SURFACES

Part of: Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  23 November 2016

GIANFRANCO CASNATI
GIANFRANCO CASNATI*
Affiliation:
Dipartimento di Scienze Matematiche, Politecnico di Torino, c.so Duca degli Abruzzi 24, 10129 Torino, Italy email [email protected]
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Abstract

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Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$ . In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces.

Keywords

vector bundleUlrich bundlestable bundledel Pezzo surface

MSC classification

Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli
Secondary: 14J45: Fano varieties
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 22 - 37
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Casanellas, M. and Hartshorne, R., ‘ACM bundles on cubic surfaces’, J. Eur. Math. Soc. 13 (2011), 709–731.Google Scholar
Casanellas, M., Hartshorne, R., Geiss, F. and Schreyer, F. O., ‘Stable Ulrich bundles’, Int. J. Math. 23 1250083 (2012), 50 pages.Google Scholar
Casnati, G., ‘On rank two aCM bundles’, Comm. Algebra (2016). Doi: 10.1080/00927872.2016.1222397.Google Scholar
Casnati, G., Faenzi, D. and Malaspina, F., ‘Rank two aCM bundles on the del Pezzo threefold with Picard number 3’, J. Algebra 429 (2015), 413–446.Google Scholar
Casnati, G., Faenzi, D. and Malaspina, F., ‘Moduli spaces of rank two aCM bundles on the Segre product of three projective lines’, J. Pure Appl. Algebra 220 (2016), 1554–1575.Google Scholar
Casnati, G. and Galluzzi, F., ‘Stability of rank $2$ Ulrich bundles on projective $K3$ surfaces’, Math. Scand., to appear.Google Scholar
Coskun, E., Kulkarni, R. S. and Mustopa, Y., ‘The geometry of Ulrich bundles on del Pezzo surfaces’, J. Algebra 375 (2013), 280–301.Google Scholar
Costa, L. and Miró-Roig, M. R., ‘On the rationality of moduli spaces of vector bundles on Fano surfaces’, J. Pure Appl. Algebra 137 (1999), 199–220.Google Scholar
Eisenbud, D. and Herzog, J., ‘The classification of homogeneous Cohen–Macaulay rings of finite representation type’, Math. Ann. 280 (1988), 347–352.Google Scholar
Eisenbud, D., Schreyer, F. O. and Weyman, J., ‘Resultants and Chow forms via exterior syzigies’, J. Amer. Math. Soc. 16 (2003), 537–579.Google Scholar
Faenzi, D., ‘Rank 2 arithmetically Cohen–Macaulay bundles on a nonsingular cubic surface’, J. Algebra 319 (2008), 143–186.Google Scholar
Faenzi, D. and Pons-Llopis, J., ‘The CM representation type of projective varieties’, Preprint, 2015, arXiv:1504.03819 [math.AG].Google Scholar
Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, 52 (Springer, New York, 1977).Google Scholar
Huybrechts, D. and Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2nd edn (Cambridge University Press, Cambridge, 2010).Google Scholar
Manin, Y., Cubic Forms (North Holland, Amsterdam, 1986).Google Scholar
Miró-Roig, R. M. and Pons-Llopis, J., ‘ n-dimensional Fano varieties of wild representation type’, J. Pure Appl. Algebra 218 (2014), 1867–1884.Google Scholar
Okonek, C., Schneider, M. and Spindler, H., Vector Bundles on Complex Projective Spaces, Progress in Mathematics, 3 (Springer, New York, 1980).Google Scholar
Pons-Llopis, J. and Tonini, F., ‘ACM bundles on del Pezzo surfaces’, Matematiche (Catania) 64 (2009), 177–211.Google Scholar
Soberon-Chavez, S., ‘Rank 2 vector bundles over a complex quadric surface’, Q. J. Math. Oxford 36 (1985), 159–172.Google Scholar