Codimension 1 distributions on three dimensional hypersurfaces. (English) Zbl 07797280
Summary: We show that codimension 1 distributions with at most isolated singularities on threefold hypersurfaces \(X_d\subset\mathbb{P}^4\) of degree \(d\) provide interesting examples of stable rank 2 reflexive sheaves. When \(d\le 5\), these sheaves can be regarded as smooth points within an irreducible component of the moduli space of stable reflexive sheaves. Our second goal goes in the reverse direction: we start from a well-known family of stable locally free sheaves and provide examples of codimension 1 distributions of local complete intersection type on \(X_d\).
MSC:
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14D22 | Fine and coarse moduli spaces |
| 14F06 | Sheaves in algebraic geometry |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
Software:
Book3264ExamplesReferences:
| [1] | O. Calvo-Andrade, M. Corrêa, and M. Jardim. Codimension One Holo-morphic Distributions on the Projective Three-space. International Math-ematics Research Notices 2020(Oct. 2018), no. 23, 9011-9074. DOI: 10. 1093/imrn/rny251. · Zbl 1460.32040 · doi:10.1093/imrn/rny251 |
| [2] | O. Calvo-Andrade, M. Corrêa, and M. Jardim. Codimension one distribu-tions and stable rank 2 reflexive sheaves on threefolds. An. Acad. Brasil. Ciênc. 93(2021), no. suppl. 3, Paper No. e20190909, 14. DOI: 10.1590/ 0001-3765202120190909. · doi:10.1590/0001-3765202120190909 |
| [3] | A. Cavalcante, M. Corrêa, and S. Marchesi. On holomorphic distributions on Fano threefolds. J. Pure Appl. Algebra 224(2020), no. 6, 106272, 20. DOI: 10.1016/j.jpaa.2019.106272. · Zbl 1439.57047 · doi:10.1016/j.jpaa.2019.106272 |
| [4] | D. Eisenbud and J. Harris. 3264 and all that-a second course in alge-braic geometry. Cambridge University Press, Cambridge, 2016, xiv+616. DOI: 10.1017/CBO9781139062046. · Zbl 1341.14001 · doi:10.1017/CBO9781139062046 |
| [5] | D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. Second. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010, xviii+325. DOI: 10.1017/CBO9780511711985. · Zbl 1206.14027 · doi:10.1017/CBO9780511711985 |
| [6] | M. Jardim. Stable bundles on 3-fold hypersurfaces. Bull. Braz. Math. Soc. (N.S.) 38(2007), no. 4, 649-659. DOI: 10 . 1007 / s00574 -007 -0067-9. · Zbl 1139.14035 · doi:10.1007/s00574-007-0067-9 |
| [7] | M. Maruyama. Moduli of stable sheaves. II. J. Math. Kyoto Univ. 18(1978), no. 3, 557-614. DOI: 10.1215/kjm/1250522511. · Zbl 0395.14006 · doi:10.1215/kjm/1250522511 |
| [8] | C. Okonek, M. Schneider, and H. Spindler. Vector bundles on complex projective spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2011, viii+239. DOI: 10.1007/978-3-0348-0151-5. · Zbl 1237.14003 · doi:10.1007/978-3-0348-0151-5 |
| [9] | G. Ottaviani. Varietá proiettive de codimension piccola. Quaderni INDAM, Aracne, Rome, 1995. URL: http://web.math.unifi.it/users /ottaviani/codim/codim.pdf. |
| [10] | T. Peternell and J. A. Wiśniewski. On stability of tangent bundles of Fano manifolds with b 2 = 1. J. Algebraic Geom. 4(1995), no. 2, 363-384. DOI: 10.48550/arXiv.alg-geom/9306010. · Zbl 0837.14033 · doi:10.48550/arXiv.alg-geom/9306010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
