Obstructions to deforming curves on a $3$-fold, I: A generalization of Mumford’s example and an application to $\operatorname {Hom}$ schemes
Authors:
Shigeru Mukai and Hirokazu Nasu
Journal:
J. Algebraic Geom. 18 (2009), 691-709
DOI:
https://doi.org/10.1090/S1056-3911-08-00502-X
Published electronically:
August 14, 2008
MathSciNet review:
2524595
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Abstract |
References |
Additional Information
Abstract: We give a sufficient condition for a first order infinitesimal deformation of a curve on a $3$-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled $3$-folds and the Hom scheme from a general curve of genus five to a general cubic $3$-fold.
References
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References
- H. Clemens and H.P. Kley: On an example of Voisin, Michigan Math. J. 48(2000), 93–119. MR 1786482 (2002c:14007)
- D. Curtin: Obstructions to deforming a space curve, Trans. Amer. Math. Soc. 267(1981), 83–94. MR 621974 (83c:14007)
- E. Dardanelli and B. van Geemen: Hessians and the moduli space of cubic surfaces, in “Algebraic Geometry” (eds. J-H. Keum and S. Kondō), Contemp. Math. 422, Amer. Math. Soc., Providence, pp. 17–36. MR 2296431 (2008d:14057)
- R. Hartshorne: Algebraic geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
- V.A. Iskovskih: Fano 3-folds. I, Math. USSR-Izv. 11(1977), 485–527 (English translation).
- J. O. Kleppe: Non-reduced components of the Hilbert scheme of smooth space curves in “Space curves” (eds. F. Ghione, C. Peskine and E. Sernesi), Lecture Notes in Math. 1266, Springer-Verlag, 1987, pp. 181–207. MR 908714 (89a:14010)
- J. Kollár: Rational curves on algebraic varieties, Springer-Verlag, 1996. MR 1440180 (98c:14001)
- D. Mumford: Further pathologies in algebraic geometry, Amer. J. Math. 84(1962), 642–648. MR 0148670 (26:6177)
- H. Nasu: Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. Res. Inst. Math. Sci. 42(2006), 117–141 (see also math.AG/0505413). MR 2215438 (2007d:14011)
- B. Segre: The non-singular cubic surfaces, Oxford University Press, Oxford, 1942. MR 0008171 (4:254b)
- R. Vakil: Murphy’s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math. 164(2006), no. 3, 569–590. MR 2227692 (2007a:14008)
Additional Information
Shigeru Mukai
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email:
mukai@kurims.kyoto-u.ac.jp
Hirokazu Nasu
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email:
nasu@kurims.kyoto-u.ac.jp
Received by editor(s):
May 23, 2007
Received by editor(s) in revised form:
September 28, 2007
Published electronically:
August 14, 2008
Additional Notes:
During this research, the first author was supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 17340006. The second author was supported in part by the 21st century COE program “Formation of an International Center of Excellence in the Frontier of Mathematics and Fostering of Researchers in Future Generations”.
