A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.
@article{bwmeta1.element.doi-10_2478_s11533-013-0308-7,
author = {Jean-Marc Dr\'ezet},
title = {Fragmented deformations of primitive multiple curves},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {2106-2137},
zbl = {1309.14023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0308-7}
}
Jean-Marc Drézet. Fragmented deformations of primitive multiple curves. Open Mathematics, Tome 11 (2013) pp. 2106-2137. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0308-7/
[1] Bănică C., Forster O., Multiplicity structures on space curves, In: The Lefschetz Centennial Conference, I, Mexico City, December 10–14, 1984, Contemp. Math., 58, American Mathematical Society, Providence, 1986, 47–64
[2] Bayer D., Eisenbud D., Ribbons and their canonical embeddings, Trans. Amer. Math. Soc., 1995, 347(3), 719–756 http://dx.doi.org/10.1090/S0002-9947-1995-1273472-3 | Zbl 0853.14016
[3] Drézet J.-M., Déformations des extensions larges de faisceaux, Pacific J. Math., 2005, 220(2), 201–297 http://dx.doi.org/10.2140/pjm.2005.220.201
[4] Drézet J.-M., Faisceaux cohérents sur les courbes multiples, Collect. Math., 2006, 57(2), 121–171
[5] Drézet J.-M., Paramétrisation des courbes multiples primitives, Adv. Geom., 2007, 7(4), 559–612 http://dx.doi.org/10.1515/ADVGEOM.2007.034
[6] Drézet J.-M., Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives, Math. Nachr., 2009, 282(7), 919–952 http://dx.doi.org/10.1002/mana.200810781
[7] Drézet J.-M., Sur les conditions d’existence des faisceaux semi-stables sur les courbes multiples primitives, Pacific J. Math., 2011, 249(2), 291–319 http://dx.doi.org/10.2140/pjm.2011.249.291
[8] Drézet J.-M., Courbes multiples primitives et déformations de courbes lisses, Ann. Fac. Sci. Toulouse Math., 2013, 22(1), 133–154 http://dx.doi.org/10.5802/afst.1368
[9] Eisenbud D., Commutative Algebra, Grad. Texts in Math., 150, Springer, Berlin-Heidelberg-New York, 1995 http://dx.doi.org/10.1007/978-1-4612-5350-1
[10] Eisenbud D., Green M., Clifford indices of ribbons, Trans. Amer. Math. Soc., 1995, 347(3), 757–765 http://dx.doi.org/10.1090/S0002-9947-1995-1273474-7 | Zbl 0854.14016
[11] González M., Smoothing of ribbons over curves, J. Reine Angew. Math., 2006, 591, 201–235 | Zbl 1094.14016
[12] Hartshorne R., Algebraic Geometry, Grad. Texts in Math., 52, Springer, Berlin-Heidelberg-New York, 1977 http://dx.doi.org/10.1007/978-1-4757-3849-0
[13] Inaba M.-A., On the moduli of stable sheaves on a reducible projective scheme and examples on a reducible quadric surface, Nagoya Math. J., 2002, 166, 135–181 | Zbl 1056.14014
[14] Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math., 1994, 79, 47–129 http://dx.doi.org/10.1007/BF02698887
[15] Teixidor i Bigas M., Moduli spaces of (semi)stable vector bundles on tree-like curves, Math. Ann., 1991, 290(2), 341–348 http://dx.doi.org/10.1007/BF01459249 | Zbl 0719.14015
[16] Teixidor i Bigas M., Moduli spaces of vector bundles on reducible curves, Amer. J. Math., 1995, 117(1), 125–139 http://dx.doi.org/10.2307/2375038 | Zbl 0836.14012
[17] Teixidor i Bigas M., Compactifications of moduli spaces of (semi)stable bundles on singular curves: two points of view, Collect. Math., 1998, 49(2–3), 527–548 | Zbl 0932.14015
[18] Zariski O., Samuel P., Commutative Algebra, I, II, Grad. Texts in Math., 28, 29, Springer, Berlin-Heidelberg-New York, 1975