On the extendability of elliptic surfaces of rank two and higher

Angelo Felice Lopez[1]; Roberto Muñoz[2]; José Carlos Sierra[3]

  • [1] Universitá di Roma Tre Dipartimento di Matematica Largo San Leonardo Murialdo 1 00146 Roma (Italy)
  • [2] Universidad Rey Juan Carlos Departamento de Matemática Aplicada 28933 Móstoles Madrid (Spain)
  • [3] Universidad Complutense de Madrid Facultad de Ciencias Matemáticas Departamento de Álgebra 28040 Madrid (Spain)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 311-346
  • ISSN: 0373-0956

Abstract

top
We study threefolds X ⊂ ℙ r having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.

How to cite

top

Lopez, Angelo Felice, Muñoz, Roberto, and Sierra, José Carlos. "On the extendability of elliptic surfaces of rank two and higher." Annales de l’institut Fourier 59.1 (2009): 311-346. <http://eudml.org/doc/10394>.

@article{Lopez2009,
abstract = {We study threefolds $X \subset \mathbb\{P\}^r$ having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.},
affiliation = {Universitá di Roma Tre Dipartimento di Matematica Largo San Leonardo Murialdo 1 00146 Roma (Italy); Universidad Rey Juan Carlos Departamento de Matemática Aplicada 28933 Móstoles Madrid (Spain); Universidad Complutense de Madrid Facultad de Ciencias Matemáticas Departamento de Álgebra 28040 Madrid (Spain)},
author = {Lopez, Angelo Felice, Muñoz, Roberto, Sierra, José Carlos},
journal = {Annales de l’institut Fourier},
keywords = {Elliptic surfaces; hyperplane sections; Mori fiber spaces; elliptic surfaces; adjunction theory},
language = {eng},
number = {1},
pages = {311-346},
publisher = {Association des Annales de l’institut Fourier},
title = {On the extendability of elliptic surfaces of rank two and higher},
url = {http://eudml.org/doc/10394},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Lopez, Angelo Felice
AU - Muñoz, Roberto
AU - Sierra, José Carlos
TI - On the extendability of elliptic surfaces of rank two and higher
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 311
EP - 346
AB - We study threefolds $X \subset \mathbb{P}^r$ having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.
LA - eng
KW - Elliptic surfaces; hyperplane sections; Mori fiber spaces; elliptic surfaces; adjunction theory
UR - http://eudml.org/doc/10394
ER -

References

top
  1. A. Altman, S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics 146 (1970), Springer-Verlag, Berlin-New York Zbl0215.37201MR274461
  2. E. Arbarello, E. Sernesi, Petri’s approach to the study of the ideal associated to a special divisor, Invent. Math. 49 (1978), 99-119 Zbl0399.14019MR511185
  3. L. Badescu, Infinitesimal deformations of negative weights and hyperplane sections, Algebraic geometry 1417 (1990), 1-22, Lecture Notes in Math. Zbl0727.14001MR1040547
  4. W. Barth, C. Peters, A. van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 4 (1984), Springer-Verlag, Berlin-New York Zbl0718.14023MR749574
  5. M. C. Beltrametti, A. J. Sommese, The adjunction theory of complex projective varieties, de Gruyter Expositions in Mathematics 16 (1995), Walter de Gruyter & Co., Berlin Zbl0845.14003MR1318687
  6. A. Bertram, L. Ein, R. Lazarsfeld, Surjectivity of Gaussian maps for line bundles of large degree on curves, Algebraic geometry 1479 (1991), 15-25, Springer, Berlin Zbl0752.14036MR1181203
  7. G. Brown, A. Corti, F. Zucconi, Birational geometry of 3-fold Mori fiber spaces, (2004), 235-275, Turin Zbl1063.14019MR2112578
  8. F. Catanese, M. Franciosi, Divisors of small genus on algebraic surfaces and projective embeddings, Israel Math. Conf. Proc. 9 (1996), 109-140, Bar-Ilan Univ., Ramat Gan Zbl0855.14004MR1360499
  9. C. Ciliberto, A. F. Lopez, R. Miranda, Projective degenerations of K 3 surfaces, Gaussian maps, and Fano threefolds, Invent. Math. 114 (1993), 641-667 Zbl0807.14028MR1244915
  10. C. Ciliberto, A. F. Lopez, R. Miranda, Classification of varieties with canonical curve section via Gaussian maps on canonical curves, Amer. J. Math. 120 (1998), 1-21 Zbl0934.14028MR1600256
  11. D. A. Cox, The Noether-Lefschetz locus of regular elliptic surfaces with section and p g ≥ 2 , Amer. J. Math. 112 (1990), 289-329 Zbl0721.14017MR1047301
  12. H. D’Souza, Threefolds whose hyperplane sections are elliptic surfaces, Pacific J. Math. 134 (1988), 57-78 Zbl0626.14031MR953500
  13. R. Elkik, Rationalité des singularités canoniques, Invent. Math. 64 (1981), 1-6 Zbl0498.14002MR621766
  14. M. L. Fania, H. D’Souza, Varieties whose surface sections are elliptic, Tohoku Math. J. 42 (1990), 457-474 Zbl0716.14022MR1076172
  15. T. Fujita, On singular del Pezzo varieties, Algebraic geometry 1417 (1990), 117-128, Springer, Berlin Zbl0714.14026MR1040555
  16. F. Gherardelli, Un teorema di Lefschetz sulle intersezioni complete, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur. 28 (1960), 610-614 Zbl0099.15802MR159827
  17. L. Giraldo, A. F. Lopez, R. Muñoz, On the existence of Enriques-Fano threefolds of index greater than one, J. Algebraic Geom. 13 (2004), 143-166 Zbl1059.14051MR2008718
  18. M. Green, Koszul cohomology and the geometry of projective varieties, J. Differ. Geom. 19 (1984), 125-171 Zbl0559.14008MR739785
  19. J. Harris, Algebraic geometry. A first course, 133 (1992), Springer-Verlag, New York Zbl0779.14001MR1182558
  20. R. Hartshorne, Algebraic geometry, 52 (1977), Springer-Verlag, New York-Heidelberg Zbl0367.14001MR463157
  21. P. Ionescu, Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986), 457-472 Zbl0619.14004MR830359
  22. V. A. Iskovskih, Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516-562, 717 Zbl0363.14010MR463151
  23. V. A. Iskovskih, Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 506-549 Zbl0407.14016MR503430
  24. Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic geometry 10 (1987), 283-360, North-Holland, Amsterdam Zbl0672.14006MR946243
  25. S. Kleiman, Les théorèmes de finitude pour le foncteur de Picard, Lecture Notes in Mathematics 225 (1971), 616-666, Springer-Verlag, Berlin-New York Zbl0227.14007
  26. R. Kloosterman, Higher Noether-Lefschetz loci of elliptic surfaces, J. Differential Geom. 76 (2007), 293-316 Zbl1141.14019MR2330416
  27. A. L. Knutsen, Smooth curves on projective K 3 surfaces, Math. Scand. 90 (2002), 215-231 Zbl1021.14013MR1895612
  28. A. L. Knutsen, A. F. Lopez, Surjectivity of Gaussian maps for curves on Enriques surfaces, Adv. Geom. 7 (2007), 215-247 Zbl1124.14035MR2314819
  29. A. L. Knutsen, A. F. Lopez, R. Muñoz, On the extendability of projective surfaces and a genus bound for Enriques-Fano threefolds, (2006) Zbl1238.14026
  30. J. Kollár, S. Mori, Birational geometry of algebraic varieties, 134 (1998), Cambridge University Press, Cambridge Zbl0926.14003MR1658959
  31. S. J. Kovács, The cone of curves of a K 3 surface, Math. Ann. 300 (1994), 681-691 Zbl0813.14026MR1314742
  32. A. Lanteri, H. Maeda, Elliptic surfaces and ample vector bundles, Pacific J. Math. 200 (2001), 147-157 Zbl1053.14051MR1863410
  33. R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (1989), 500-559, World Sci. Publishing, Teaneck, NJ Zbl0800.14003MR1082360
  34. R. Lazarsfeld, Positivity in Algebraic geometry. I, 48 (2004), Springer-Verlag, Berlin Zbl1093.14500MR2095471
  35. A. F. Lopez, Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc. 89 (1991) Zbl0736.14012MR1043786
  36. S. L’vovsky, Extensions of projective varieties and deformations. I, Michigan Math. J. 39 (1992), 41-51 Zbl0770.14005MR1137887
  37. K. Matsuki, Introduction to the Mori program, (2002), Springer-Verlag, New York Zbl0988.14007MR1875410
  38. R. Miranda, The moduli of Weierstrass fibrations over ℙ 1 , Math. Ann. 255 (1981), 379-394 Zbl0438.14023MR615858
  39. R. Miranda, The basic theory of elliptic surfaces, (1989), ETS Editrice, Pisa Zbl0744.14026MR1078016
  40. B. G. Moishezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225-268 Zbl0162.52503MR213351
  41. S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982), 133-176 Zbl0557.14021MR662120
  42. S. Mori, S. Mukai, Classification of Fano 3 -folds with B 2 ≥ 2 , Manuscripta Math. 36 (1981/82), 147-162 Zbl0478.14033MR641971
  43. M. Nagata, On rational surfaces. I. Irreducible curves of arithmetic genus 0 or 1 , Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351-370 Zbl0100.16703MR126443
  44. Y. Namikawa, Smoothing Fano 3-folds, J. Algebraic Geom. 6 (1997), 307-324 Zbl0906.14019MR1489117
  45. R. Paoletti, Free pencils on divisors, Math. Ann. 303 (1995), 109-123 Zbl0835.14005MR1348358
  46. Yu. G. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Mat. Sb. 196 (2005), 81-122 Zbl1081.14058MR2141325
  47. V. V. Shokurov, The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 922-964, 968 Zbl0422.14019MR548510
  48. V. V. Shokurov, Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 430-441 Zbl0407.14017MR534602
  49. J. Wahl, Introduction to Gaussian maps on an algebraic curve, Complex Projective Geometry. Trieste-Bergen 1989, Lond. Math. Soc. Lect. Note Ser. 179, Cambridge Univ. Press, Cambridge 1992, 304-323 Zbl0790.14014
  50. F. L. Zak, Some properties of dual varieties and their applications in projective geometry, Algebraic geometry 1479 (1991), 273-280, Springer, Berlin Zbl0793.14026MR1181218

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.