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Thiago Fassarella
Foliations with Degenerate Gauss maps on $\mathbb{P}^4$
(Feuilletages avec application de Gauss dégénérée sur $\mathbb{P}^4$)
Annales de l'institut Fourier, 60 no. 2 (2010), p. 455-487, doi: 10.5802/aif.2529
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2667783 | Zbl 1192.37067
Class. Math.: 37F75, 32M25, 34M45
Keywords: Gauss Map, Degenerate, Holomorphic Foliations
[1] B. Adlandsvik, “Joins and Higher secant varieties”, Math. Scand. 61 (1987), p. 213-222 MR 947474 | Zbl 0657.14034
[2] M.A. Akivis & V.V. Gol’dberg, Differential geometry of varieties with degenerate Gauss maps, Springer, 2004 MR 2014407
[3] Marco Brunella, Birational geometry of foliations, Monografias de Matemática. [Mathematical Monographs], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000 MR 1948251 | Zbl 1073.14022
[4] D. Cerveau & A. Lins Neto, “Irreducible Components of the Space of Holomorphic Foliations of Degree Two in CP(n), n $\ge $ 3”, The Annals of Mathematics 143 (1996) no. 3, p. 577-612
Article | MR 1394970 | Zbl 0855.32015
[5] D. Cerveau, A.L. Neto, F. Loray, J.V. Pereira & F. Touzet, “Algebraic Reduction Theorem for complex codimension one singular foliations”, Comment. Math. Helv. 81 (2006) no. 1, p. 157-169
Article | MR 2208802 | Zbl 1095.37019
[6] Dominique Cerveau, “Feuilletages en droites, équations des eikonales et aures équations différentielles”, arXiv:math.DS/0505601v1, 2005
arXiv | MR 1760842
[7] M. Dale, “Terracini’s lemma and the secant variety of a curve”, Proc. London Math. Soc. 3 (1984), p. 329-339
Article | MR 748993 | Zbl 0571.14025
[8] P. De Poi, “On first order congruences of lines of $\mathbb{P}^4$ with a fundamental curve”, manuscripta mathematica 106 (2001) no. 1, p. 101-116
Article | MR 1860982 | Zbl 1066.14062
[9] P. De Poi, “Congruences of lines with one-dimensional focal locus”, Portugaliae Mathematica 61 (2004) no. 3, p. 329-338 MR 2098024 | Zbl 1067.14049
[10] P. De Poi, “On first order congruences of lines in $\mathbb{P}^4$ with irreducible fundamental surface”, Mathematische Nachrichten 278 (2005) no. 4, p. 363-378
Article | MR 2121565 | Zbl 1070.14045
[11] P. De Poi, “On First Order Congruences of Lines in $\mathbb{P}^4$ with Generically Non-reduced Fundamental Surface”, Asian Journal of Mathematics 12 (2008), p. 56-64
Article | MR 2415011 | Zbl 1147.14025
[12] T. Fassarella & J.V. Pereira, “On the degree of polar transformations. An approach through logarithmic foliations”, Selecta Mathematica, New Series 13 (2007) no. 2, p. 239-252
Article | MR 2361094 | Zbl pre05242806
[13] G. Fischer & J. Piontkowski, Ruled varieties: an introduction to algebraic differential geometry, Vieweg Verlag, 2001 MR 1876644 | Zbl 0976.14025
[14] P. Griffiths & J. Harris, “Algebraic geometry and local differential geometry”, Ann. Sci. Ecole Norm. Sup.(4) 12 (1979) no. 3, p. 355-452
Numdam | MR 559347 | Zbl 0426.14019
[15] T.A. Ivey & J.M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, American Mathematical Society, 2003 MR 2003610 | Zbl 1105.53001
[16] JP Jouanolou, “Equations de Pfaff algebriques, in Lectures Notes in Mathematics, 708”, 1979 MR 537038 | Zbl 0477.58002
[17] EE Kummer, “Über die algebraischen Strahlensysteme, insbesondere über die der ersten und zweiten Ordnung”, Abh. K. Preuss. Akad. Wiss. Berlin (1866), p. 1-120, also in EE Kummer, Collected Papers, Springer Verlag, 1975
[18] G. Marletta, “Sopra i complessi d ordine uno dell $S^4$, Atti Accad”, Gioenia, Serie V, Catania 3 (1909), p. 1-15 Zbl 0020.39301
[19] G. Marletta, “Sui complessi di rette del primo ordine dello spazio a quattro dimensioni”, Rend. Circ. Mat. Palermo 28 (1909), p. 353-399
Article | JFM 40.0723.01
[20] E. Mezzetti & D. Portelli, “A tour through some classical theorems on algebraic surfaces”, An. Stiint. Univ. Ovidius Constanta Ser. Mat 5 (1997), p. 51-78 MR 1614780 | Zbl 0971.14032
[21] E. Mezzetti & O. Tommasi, “On projective varieties of dimension $n+k$ covered by $k$-spaces”, Illinois J.Math. 46 (2002) no. 2, p. 443-465
Article | MR 1936928 | Zbl 1052.14065
[22] JV Pereira & S. Yuzvinsky, “Completely reducible hypersurfaces in a pencil”, Advances in Mathematics 219 (2008) no. 2, p. 672-688
Article | MR 2435653 | Zbl 1146.14005
[23] Z. Ran, “Surfaces of order 1 in Grassmannians”, J. reine angew. Math 368 (1986), p. 119-126
Article | MR 850617 | Zbl 0601.14042
[24] E. Rogora, “Classification of Bertini’s series of varieties of dimension less than or equal to four”, Geometriae Dedicata 64 (1997) no. 2, p. 157-191
Article | MR 1436764 | Zbl 0893.14019
[25] C. Segre, “Su una classe di superficie degl’iperspazii legate colle equazioni lineari alle derivate parziali di 2 ordine”, Atti della R. Accademia delle Scienze di Torino 42 (1906), p. 559-591 JFM 38.0671.04
[26] C. Segre, “Preliminari di una teoria delle varieta luoghi di spazi”, Rendiconti del Circolo Matematico di Palermo 30 (1910) no. 1, p. 87-121
Article | JFM 41.0724.01
[27] C. Segre, “Le superficie degli iperspazi con una doppia infinita di curve piane o spaziali”, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur 56 (1920), p. 75-89 JFM 48.0764.02
[28] A. Terracini, “Sulle $V_k$ per cui la varieta degli $S_h$ $(h+1)$–seganti ha dimensione minore dell’ ordinario”, Rend. Circ. Mat. Palermo 31 (1911), p. 392-396
Article | JFM 42.0673.02
[29] F. Zak, Tangents and secants of algebraic varieties, Translations of mathematical monographs 127, American Mathematical Society, 1993 MR 1234494 | Zbl 0795.14018
Thiago Fassarella
Foliations with Degenerate Gauss maps on $\mathbb{P}^4$
(Feuilletages avec application de Gauss dégénérée sur $\mathbb{P}^4$)
Annales de l'institut Fourier, 60 no. 2 (2010), p. 455-487, doi: 10.5802/aif.2529
Article: subscription required (your ip address: 107.21.156.140) | Reviews MR 2667783 | Zbl 1192.37067
Class. Math.: 37F75, 32M25, 34M45
Keywords: Gauss Map, Degenerate, Holomorphic Foliations
Résumé - Abstract
We obtain a classification of codimension one holomorphic foliations on $\mathbb{P}^4$ with degenerate Gauss maps.
Bibliography
[2] M.A. Akivis & V.V. Gol’dberg, Differential geometry of varieties with degenerate Gauss maps, Springer, 2004 MR 2014407
[3] Marco Brunella, Birational geometry of foliations, Monografias de Matemática. [Mathematical Monographs], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000 MR 1948251 | Zbl 1073.14022
[4] D. Cerveau & A. Lins Neto, “Irreducible Components of the Space of Holomorphic Foliations of Degree Two in CP(n), n $\ge $ 3”, The Annals of Mathematics 143 (1996) no. 3, p. 577-612
Article | MR 1394970 | Zbl 0855.32015
[5] D. Cerveau, A.L. Neto, F. Loray, J.V. Pereira & F. Touzet, “Algebraic Reduction Theorem for complex codimension one singular foliations”, Comment. Math. Helv. 81 (2006) no. 1, p. 157-169
Article | MR 2208802 | Zbl 1095.37019
[6] Dominique Cerveau, “Feuilletages en droites, équations des eikonales et aures équations différentielles”, arXiv:math.DS/0505601v1, 2005
arXiv | MR 1760842
[7] M. Dale, “Terracini’s lemma and the secant variety of a curve”, Proc. London Math. Soc. 3 (1984), p. 329-339
Article | MR 748993 | Zbl 0571.14025
[8] P. De Poi, “On first order congruences of lines of $\mathbb{P}^4$ with a fundamental curve”, manuscripta mathematica 106 (2001) no. 1, p. 101-116
Article | MR 1860982 | Zbl 1066.14062
[9] P. De Poi, “Congruences of lines with one-dimensional focal locus”, Portugaliae Mathematica 61 (2004) no. 3, p. 329-338 MR 2098024 | Zbl 1067.14049
[10] P. De Poi, “On first order congruences of lines in $\mathbb{P}^4$ with irreducible fundamental surface”, Mathematische Nachrichten 278 (2005) no. 4, p. 363-378
Article | MR 2121565 | Zbl 1070.14045
[11] P. De Poi, “On First Order Congruences of Lines in $\mathbb{P}^4$ with Generically Non-reduced Fundamental Surface”, Asian Journal of Mathematics 12 (2008), p. 56-64
Article | MR 2415011 | Zbl 1147.14025
[12] T. Fassarella & J.V. Pereira, “On the degree of polar transformations. An approach through logarithmic foliations”, Selecta Mathematica, New Series 13 (2007) no. 2, p. 239-252
Article | MR 2361094 | Zbl pre05242806
[13] G. Fischer & J. Piontkowski, Ruled varieties: an introduction to algebraic differential geometry, Vieweg Verlag, 2001 MR 1876644 | Zbl 0976.14025
[14] P. Griffiths & J. Harris, “Algebraic geometry and local differential geometry”, Ann. Sci. Ecole Norm. Sup.(4) 12 (1979) no. 3, p. 355-452
Numdam | MR 559347 | Zbl 0426.14019
[15] T.A. Ivey & J.M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, American Mathematical Society, 2003 MR 2003610 | Zbl 1105.53001
[16] JP Jouanolou, “Equations de Pfaff algebriques, in Lectures Notes in Mathematics, 708”, 1979 MR 537038 | Zbl 0477.58002
[17] EE Kummer, “Über die algebraischen Strahlensysteme, insbesondere über die der ersten und zweiten Ordnung”, Abh. K. Preuss. Akad. Wiss. Berlin (1866), p. 1-120, also in EE Kummer, Collected Papers, Springer Verlag, 1975
[18] G. Marletta, “Sopra i complessi d ordine uno dell $S^4$, Atti Accad”, Gioenia, Serie V, Catania 3 (1909), p. 1-15 Zbl 0020.39301
[19] G. Marletta, “Sui complessi di rette del primo ordine dello spazio a quattro dimensioni”, Rend. Circ. Mat. Palermo 28 (1909), p. 353-399
Article | JFM 40.0723.01
[20] E. Mezzetti & D. Portelli, “A tour through some classical theorems on algebraic surfaces”, An. Stiint. Univ. Ovidius Constanta Ser. Mat 5 (1997), p. 51-78 MR 1614780 | Zbl 0971.14032
[21] E. Mezzetti & O. Tommasi, “On projective varieties of dimension $n+k$ covered by $k$-spaces”, Illinois J.Math. 46 (2002) no. 2, p. 443-465
Article | MR 1936928 | Zbl 1052.14065
[22] JV Pereira & S. Yuzvinsky, “Completely reducible hypersurfaces in a pencil”, Advances in Mathematics 219 (2008) no. 2, p. 672-688
Article | MR 2435653 | Zbl 1146.14005
[23] Z. Ran, “Surfaces of order 1 in Grassmannians”, J. reine angew. Math 368 (1986), p. 119-126
Article | MR 850617 | Zbl 0601.14042
[24] E. Rogora, “Classification of Bertini’s series of varieties of dimension less than or equal to four”, Geometriae Dedicata 64 (1997) no. 2, p. 157-191
Article | MR 1436764 | Zbl 0893.14019
[25] C. Segre, “Su una classe di superficie degl’iperspazii legate colle equazioni lineari alle derivate parziali di 2 ordine”, Atti della R. Accademia delle Scienze di Torino 42 (1906), p. 559-591 JFM 38.0671.04
[26] C. Segre, “Preliminari di una teoria delle varieta luoghi di spazi”, Rendiconti del Circolo Matematico di Palermo 30 (1910) no. 1, p. 87-121
Article | JFM 41.0724.01
[27] C. Segre, “Le superficie degli iperspazi con una doppia infinita di curve piane o spaziali”, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur 56 (1920), p. 75-89 JFM 48.0764.02
[28] A. Terracini, “Sulle $V_k$ per cui la varieta degli $S_h$ $(h+1)$–seganti ha dimensione minore dell’ ordinario”, Rend. Circ. Mat. Palermo 31 (1911), p. 392-396
Article | JFM 42.0673.02
[29] F. Zak, Tangents and secants of algebraic varieties, Translations of mathematical monographs 127, American Mathematical Society, 1993 MR 1234494 | Zbl 0795.14018
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310