Logarithmic vector bundles on the blown-up variety. (English) Zbl 1542.14020
Given a divisor \(D\subset X\) on a smooth projective variety, Deligne defined the logarithmic sheaf \(\Omega^1_X(\log D)\) of differential \(1\)-forms with logarithmic poles along \(D\). Among many other problems related with logarithmic sheaves, the “Torelli problem” of finding out in which cases the divisor \(D\subset X\) can be recovered from its associated logarithmic sheaf \(\Omega^1_X(\log D)\) has raised a lot of attention. In this paper the authors approach this problem in the case of the blow-up of a variety along a finite set of points and obtain a full answer in the particular case of the del Pezzo surface \(X\subset \mathbb{P}^8\) of degree \(8\) for divisors consisting of arrangements of rational curves of degree \(3\).
Reviewer: Joan Pons-Llopis (Maó)
MSC:
| 14F06 | Sheaves in algebraic geometry |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14C34 | Torelli problem |
References:
| [1] | Angelini, E., Logarithmic bundles of hypersurface arrangements in \(\mathbb{P}^n \), Collect. Math., 65, 3, 285-302, 2014 · Zbl 1315.14055 · doi:10.1007/s13348-014-0112-0 |
| [2] | Angelini, E., Logarithmic bundles of multi-degree arrangements in \(\mathbb{P}^n \), Doc. Math., 20, 507-529, 2015 · Zbl 1348.14130 · doi:10.4171/dm/497 |
| [3] | Aprodu, M.; Brînzănescu, V.; Marchitan, M., Rank-two vector bundles on Hirzebruch surfaces, Open Math., 10, 4, 1321-1330, 2012 · Zbl 1282.14071 |
| [4] | Ballico, E.; Huh, S.; Malaspina, F., A Torelli-type problem for logarithmic bundles over projective varieties, Q. J. Math., 66, 2, 417-436, 2015 · Zbl 1349.14033 · doi:10.1093/qmath/hau034 |
| [5] | Brînzânescu, V., Algebraic 2-vector bundles on ruled surfaces, Annali dell’Università di Ferrara, 37, 1, 55-64, 1991 · Zbl 0795.14010 · doi:10.1007/BF02825275 |
| [6] | Brosius, JE, Rank-2 vector bundles on a ruled surface, I, Mathematische Annalen, 265, 2, 155-168, 1983 · Zbl 0503.55012 · doi:10.1007/BF01460796 |
| [7] | Deligne, P., Théorie de hodge: II, Publ Math l’IHÉS, 40, 5-57, 1971 · Zbl 0219.14007 · doi:10.1007/BF02684692 |
| [8] | Deligne, P., Équations Différentielles à Points Singuliers Réguliers, 1970, Berlin: Springer-Verlag, Berlin · Zbl 0244.14004 · doi:10.1007/BFb0061194 |
| [9] | Dolgachev, I.; Kapranov, M., Arrangements of hyperplanes and vector bundles on \(\mathbb{P}^n\), Duke Math. J., 71, 3, 633-664, 1993 · Zbl 0804.14007 · doi:10.1215/S0012-7094-93-07125-6 |
| [10] | Friedman, R., Algebraic Surfaces and Holomorphic Vector Bundles, 1998, New York: Springer-Verlag, New York · Zbl 0902.14029 · doi:10.1007/978-1-4612-1688-9 |
| [11] | Fulton, W., Intersectoin Theory, 2013, Berlin: Springer, Berlin |
| [12] | Hartshorne, R., Algebraic Geometry, 1977, New York: Springer-Verlag, New York · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0 |
| [13] | Huh, S.; Marchesi, S.; Pons-Llopis, J.; Vallés, J., Generalized logarithmic sheaf on smooth projective surfaces, Int. Math. Res. Not., 2023, 21, 18387-18442, 2023 · Zbl 07794973 · doi:10.1093/imrn/rnad029 |
| [14] | Huybrechts, D.; Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2010, Cambridge: Cambridge University Press, Cambridge · Zbl 1206.14027 · doi:10.1017/CBO9780511711985 |
| [15] | Ishmura, S., On \(\pi \)-uniform vector bundles, Tokyo J. Math., 2, 2, 337-342, 1979 · Zbl 0422.14009 |
| [16] | Ueda, K.; Yoshinaga, M., Logarithmic vector fields along smooth divisors in projective spaces, Hokkaido Math. J., 38, 3, 409-415, 2009 · Zbl 1180.14012 · doi:10.14492/hokmj/1258553970 |
| [17] | Okonek, C.; Michael, S.; Heinz, S.; Gelfand, SI, Vector Bundles on Complex Projective Spaces, 1980, Berlin: Springer, Berlin · doi:10.1007/978-3-0348-0151-5 |
| [18] | Vallès, J., Nombre maximal d’hyperplans instables pour un fibré de Steiner, Math. Z., 233, 3, 507-514, 2000 · Zbl 0952.14011 · doi:10.1007/s002090050484 |
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