Abstract
Hilbert schemes of suitable smooth, projective threefold scrolls over the Hirzebruch surface 𝔽e, e ≥ 2, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension, and the general point of such a component is described.
Communicated by: A. Sommese
Acknowledgements
The authors thank C. Ciliberto and E. Sernesi for having pointed out questions on Hilbert schemes of threefold scrolls over 𝔽e, with e ≥ 2, during the talk of the first author at the Workshop “Algebraic geometry: two days in Rome two”, held in Rome in February 2012. The authors are also grateful to the referee for helpful comments and for having posed a question which allowed us to realize that there was a mistake in the first version of the paper. Both authors are members of GNSAGA-INdAM.
Funding: We acknowledge partial support from MIUR funds, PRIN 2010-2011 project “Geometria delle Varietà Algebriche”.
References
[1] A. Alzati, G. M. Besana, Criteria for very ampleness of rank two vector bundles over ruled surfaces. Canad.J. Math. 62 (2010), 1201–1227. MR2760655 Zbl 1209.1401310.4153/CJM-2010-066-5Search in Google Scholar
[2] M. Aprodu, V. Brínzănescu, Moduli spaces of vector bundles over ruled surfaces. Nagoya Math.J. 154 (1999), 111–122. MR1689175 Zbl 0938.1402410.1017/S0027763000025332Search in Google Scholar
[3] E. Arrondo, M. Pedreira, I. Sols, On regular and stable ruled surfaces in P3. In: Algebraic curves and projective geometry (Trento, 1988), volume 1389 of Lecture Notes in Math., 1–15, Springer 1989. MR1023385 Zbl 0694.1401510.1007/BFb0085919Search in Google Scholar
[4] E. Ballico, Coherent sheaves on ℙ1: their families and their deformations related to a behavioral approach to singular systems. ActaAppl. Math. 66 (2001), 123–138. MR1837616 Zbl 1065.1450410.1023/A:1010628724686Search in Google Scholar
[5] M. C. Beltrametti, A. J. Sommese, The adjunction theory of complex projective varieties. De Gruyter 1995. MR1318687 Zbl 0845.1400310.1515/9783110871746Search in Google Scholar
[6] G. M. Besana, M. L. Fania, The dimension of the Hilbert scheme of special threefolds. Comm. Algebra 33 (2005), 3811–3829. MR2175469 Zbl 1093.1405810.1080/00927870500242926Search in Google Scholar
[7] G. M. Besana, M. L. Fania, F. Flamini, Hilbert scheme of some threefold scrolls over the Hirzebruch surface F1. J. Math. Soc. Japan 65 (2013), 1243–1272. MR3127823 Zbl 1284.1405010.2969/jmsj/06541243Search in Google Scholar
[8] F. A. Bogomolov, Stable vector bundles on projective surfaces. Mat. Sb. 185 (1994), 3–26. MR1272185 Zbl 0838.1403610.1070/SM1995v081n02ABEH003544Search in Google Scholar
[9] J. E. Brosius, Rank-2 vector bundles on a ruled surface. I. Math. Ann. 265 (1983), 155–168. MR719134 Zbl 0503.5501210.1007/BF01460796Search in Google Scholar
[10] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, Degenerations of scrolls to unions of planes. AttiAccad. Naz. Lincei Rend. Lincei Mat. Appl. 17 (2006), 95–123. MR2238370 Zbl 1136.1402510.4171/RLM/457Search in Google Scholar
[11] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, Non-special scrolls with general moduli. Rend. Circ. Mat. Palermo (2) 57 (2008), 1–31. MR2420521 Zbl 1222.1408210.1007/s12215-008-0001-zSearch in Google Scholar
[12] A. Calabri, C. Ciliberto, F. Flamini, R. Miranda, Special scrolls whose base curve has general moduli. In: Interactions of classical and numerical algebraic geometry, volume 496 of Contemp. Math., 133–155, Amer. Math. Soc. 2009. MR2555952 Zbl 1183.1404910.1090/conm/496/09721Search in Google Scholar
[13] M.-C. Chang, The number of components of Hilbert schemes. Internat. J. Math. 7 (1996), 301–306. MR1395932 Zbl 0892.1400610.1142/S0129167X96000189Search in Google Scholar
[14] M.-C. Chang, Inequidimensionality of Hilbert schemes. Proc. Amer. Math. Soc. 125 (1997), 2521–2526. MR1389509 Zbl 0883.1400110.1090/S0002-9939-97-03836-7Search in Google Scholar
[15] C. Ciliberto, F. Flamini, Brill-Noether loci of stable rank-two vector bundles on a general curve. In: Geometry and arithmetic, 61–74, Eur. Math. Soc., Zürich 2012. MR2987653 Zbl 1317.1407510.4171/119-1/4Search in Google Scholar
[16] C. Ciliberto, A. Lopez, R. Miranda, Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds. Invent. Math. 114 (1993), 641–667. MR1244915 Zbl 0807.1402810.1007/BF01232682Search in Google Scholar
[17] G. Ellingsrud, Sur le schèma de Hilbert des variétés de codimension 2 dans ℙe à cône de Cohen-Macaulay. Ann. Sci. École Norm. Sup. (4) 8 (1975), 423–431. MR0393020 Zbl 0325.1400210.24033/asens.1297Search in Google Scholar
[18] D. Faenzi, M. L. Fania, Skew-symmetric matrices and Palatini scrolls. Math. Ann. 347 (2010), 859–883. MR2658146 Zbl 1200.1402910.1007/s00208-009-0450-5Search in Google Scholar
[19] D. Faenzi, M. L. Fania, On the Hilbert scheme of varieties defined by maximal minors. Math. Res. Lett. 21 (2014), 297–311. MR3247058 Zbl 1304.1406310.4310/MRL.2014.v21.n2.a8Search in Google Scholar
[20] M. L. Fania, E. Mezzetti, On the Hilbert scheme of Palatini threefolds. Adv. Geom. 2 (2002), 371–389. MR1940444 Zbl 1054.1405210.1515/advg.2002.017Search in Google Scholar
[21] F. Flamini, Pr-scrolls arising from Brill-Noether theory and K3-surfaces. Manuscripta Math. 132 (2010), 199–220. MR2609294 Zbl 1194.1405810.1007/s00229-010-0343-7Search in Google Scholar
[22] R. Friedman, Algebraic surfaces and holomorphic vector bundles. Springer 1998. MR1600388 Zbl 0902.1402910.1007/978-1-4612-1688-9Search in Google Scholar
[23] R. Friedman, D. R. Morrison, The birational geometry of degenerations: an overview. In: The birationalgeometry of degenerations (Cambridge, Mass., 1981), 1–32, Birkhäuser 1983. MR690262 Zbl 0508.14024Search in Google Scholar
[24] L. Fuentes García, M. Pedreira, Canonical geometrically ruled surfaces. Math. Nachr. 278 (2005), 240–257. MR2110530 Zbl 1067.1403010.1002/mana.200310238Search in Google Scholar
[25] L. Fuentes García, M. Pedreira Pérez, The generic special scroll of genus g in ℙN: special scrolls in ℙ3. Commun. Algebra 40 (2012), 4483–4493. Zbl 1272.1402710.1080/00927872.2011.610210Search in Google Scholar
[26] W. Fulton, Intersection theory. Springer 1984. MR732620 Zbl 0541.1400510.1007/978-3-662-02421-8Search in Google Scholar
[27] F. Ghione, Quelques résultats de Corrado Segre sur les surfaces règlèes. Math. Ann. 255 (1981), 77–95. MR611274 Zbl 0435.1401010.1007/BF01450557Search in Google Scholar
[28] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert. In: Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249–276, Soc. Math. France, Paris 1995. MR1611822 Zbl 0142.33504Search in Google Scholar
[29] R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.1400110.1007/978-1-4757-3849-0Search in Google Scholar
[30] D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves. Friedr. Vieweg & Sohn, Braunschweig 1997. MR1450870 Zbl 0872.1400210.1007/978-3-663-11624-0Search in Google Scholar
[31] J. O. Kleppe, Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes. J. Algebra 407 (2014), 246 -274. MR3197160 Zbl 1328.1408010.1016/j.jalgebra.2014.03.007Search in Google Scholar
[32] J. O. Kleppe, J. C. Migliore, R. Miró-Roig, U. Nagel, C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness. Mem. Amer. Math. Soc. 154 (2001), viii+116. MR1848976 Zbl 1006.1401810.1090/memo/0732Search in Google Scholar
[33] J. O. Kleppe, R. M. Miró-Roig, The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes. J. Pure Appl. Algebra 127 (1998), 73–82. MR1609504 Zbl 0949.1400310.1016/S0022-4049(96)00180-6Search in Google Scholar
[34] M. Maruyama, On automorphism groups of ruled surfaces. J. Math. Kyoto Univ. 11 (1971), 89–112. MR0280493 Zbl 0213.4780310.1215/kjm/1250523688Search in Google Scholar
[35] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces. Birkhäuser 1980. MR561910 Zbl 0438.3201610.1007/978-3-0348-0151-5Search in Google Scholar
[36] G. Ottaviani, On 3-folds in ℙ5 which are scrolls. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 451–471. MR1205407 Zbl 0786.14026Search in Google Scholar
[37] C. Segre, Recherches générales sur les sourbes et les surfaces réglées algébriques. Math. Ann. 34 (1889), 1–25. MR1510566 Zbl 19.0676.0210.1007/BF01446790Search in Google Scholar
[38] E. Sernesi, Deformations of algebraic schemes. Springer 2006. MR2247603 Zbl 1102.14001Search in Google Scholar
[39] A. J. Sommese, On the minimality of hyperplane sections of projective threefolds. J. Reine Angew. Math. 329 (1981), 16–41. MR636441 Zbl 0509.14044Search in Google Scholar
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