Quartic del Pezzo surfaces over function fields of curves

Brendan Hassett; Yuri Tschinkel

Open Mathematics (2014)

  • Volume: 12, Issue: 3, page 395-420
  • ISSN: 2391-5455

Abstract

top
We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

How to cite

top

Brendan Hassett, and Yuri Tschinkel. "Quartic del Pezzo surfaces over function fields of curves." Open Mathematics 12.3 (2014): 395-420. <http://eudml.org/doc/269089>.

@article{BrendanHassett2014,
abstract = {We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.},
author = {Brendan Hassett, Yuri Tschinkel},
journal = {Open Mathematics},
keywords = {Del Pezzo surfaces; Fibrations; Function fields of curves; Rational points; Intermediate Jacobians; del Pezzo surfaces; fibrations; function fields of curves; rational points; intermediate Jacobians},
language = {eng},
number = {3},
pages = {395-420},
title = {Quartic del Pezzo surfaces over function fields of curves},
url = {http://eudml.org/doc/269089},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Brendan Hassett
AU - Yuri Tschinkel
TI - Quartic del Pezzo surfaces over function fields of curves
JO - Open Mathematics
PY - 2014
VL - 12
IS - 3
SP - 395
EP - 420
AB - We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.
LA - eng
KW - Del Pezzo surfaces; Fibrations; Function fields of curves; Rational points; Intermediate Jacobians; del Pezzo surfaces; fibrations; function fields of curves; rational points; intermediate Jacobians
UR - http://eudml.org/doc/269089
ER -

References

top
  1. [1] Baker H.F., On the invariants of a binary quintic and the reality of its roots, Proc. London Math. Soc., 1908, s2–6(1), 122–140 http://dx.doi.org/10.1112/plms/s2-6.1.122 Zbl39.0152.04
  2. [2] Beauville A., Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup., 1977, 10(3), 309–391 Zbl0368.14018
  3. [3] Brumer A., Remarques sur les couples de formes quadratiques, C. R. Acad. Sci. Paris Sér. A-B, 1978, 286(16), A679–A681 Zbl0392.10021
  4. [4] Castravet A.-M., Rational families of vector bundles on curves, Internat. J. Math., 2004, 15(1), 13–45 http://dx.doi.org/10.1142/S0129167X0400220X Zbl1092.14041
  5. [5] Cheltsov I., Nonrational nodal quartic threefolds, Pacific J. Math., 2006, 226(1), 65–81 http://dx.doi.org/10.2140/pjm.2006.226.65 Zbl1123.14010
  6. [6] Debarre O., Iliev A., Manivel L., On nodal prime Fano threefolds of degree 10, Sci. China Math., 2011, 54(8), 1591–1609 http://dx.doi.org/10.1007/s11425-011-4182-0 Zbl1247.14043
  7. [7] Debarre O., Iliev A., Manivel L., On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom., 2012, 21(1), 21–59 http://dx.doi.org/10.1090/S1056-3911-2011-00594-8 Zbl1250.14029
  8. [8] Deligne P., Illusie L., Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., 1987, 89(2), 247–270 http://dx.doi.org/10.1007/BF01389078 Zbl0632.14017
  9. [9] Donagi R., The tetragonal construction, Bull. Amer. Math. Soc. (N.S.), 1981, 4(2), 181–185 http://dx.doi.org/10.1090/S0273-0979-1981-14875-8 Zbl0491.14016
  10. [10] Esnault H., Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math., 2003, 151(1), 187–191 http://dx.doi.org/10.1007/s00222-002-0261-8 Zbl1092.14010
  11. [11] Graber T., Harris J., Starr J., Families of rationally connected varieties, J. Amer. Math. Soc., 2003, 16(1), 57–67 http://dx.doi.org/10.1090/S0894-0347-02-00402-2 Zbl1092.14063
  12. [12] Harris J., Roth M., Starr J., Curves of small degree on cubic threefolds, Rocky Mountain J. Math., 2005, 35(3), 761–817 http://dx.doi.org/10.1216/rmjm/1181069707 Zbl1080.14008
  13. [13] Harris J., Roth M., Starr J., Abel-Jacobi maps associated to smooth cubic threefolds, preprint available at http://arxiv.org/abs/math/0202080 
  14. [14] Hassett B., Rational surfaces over nonclosed fields, In: Arithmetic Geometry, Clay Math. Proc., 8, American Mathematical Society, Providence, 2009, 155–209 Zbl1191.14046
  15. [15] Hassett B., Hyeon D., Log minimal model program for the moduli space of stable curves: the first flip, Ann. of Math., 2013, 177(3), 911–968 http://dx.doi.org/10.4007/annals.2013.177.3.3 Zbl1273.14034
  16. [16] Hassett B., Tschinkel Yu., uEmbedding pointed curves in K3 surfaces, preprint available at http://arxiv.org/abs/1301.7262 
  17. [17] Hassett B., Tschinkel Yu., Spaces of sections of quadric surface fibrations over curves, In: Compact Moduli Spaces and Vector Bundles, Contemp. Math., 564, American Mathematical Society, Providence, 2012, 227–249 http://dx.doi.org/10.1090/conm/564/11161 
  18. [18] Hosoh T., Automorphism groups of quartic del Pezzo surfaces, J. Algebra, 1996, 185(2), 374–389 http://dx.doi.org/10.1006/jabr.1996.0331 
  19. [19] Iliev A., Markushevich D., The Abel-Jacobi map for a cubic threefold and periods of Fano threefolds of degree 14, Doc. Math., 2000, 5, 23–47 Zbl0938.14021
  20. [20] Iliev A., Markushevich D., atParametrization of sing Θ for a Fano 3-fold of genus 7 by moduli of vector bundles, Asian J. Math., 2007, 11(3), 427–458 http://dx.doi.org/10.4310/AJM.2007.v11.n3.a4 Zbl1136.14031
  21. [21] de Jong A.J., Starr J., Every rationally connected variety over the function field of a curve has a rational point, Amer. J. Math., 2003, 125(3), 567–580 http://dx.doi.org/10.1353/ajm.2003.0017 Zbl1063.14025
  22. [22] Kanev V., Intermediate Jacobians and Chow groups of threefolds with a pencil of del Pezzo surfaces, Ann. Mat. Pura Appl., 1989, 154, 13–48 http://dx.doi.org/10.1007/BF01790341 Zbl0708.14030
  23. [23] Koitabashi M., Automorphism groups of generic rational surfaces, J. Algebra, 1988, 116(1), 130–142 http://dx.doi.org/10.1016/0021-8693(88)90196-2 
  24. [24] Kollár J., Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb., 32, Springer, Berlin, 1996 http://dx.doi.org/10.1007/978-3-662-03276-3 
  25. [25] Lang S., atAbelian varieties over finite fields, Proc. Nat. Acad. Sci. U.S.A., 1955, 41, 174–176 http://dx.doi.org/10.1073/pnas.41.3.174 
  26. [26] Leep D.B., The Amer-Brumer theorem over arbitrary fields, preprint available at www.ms.uky.edu/∼leep/Preprints.html 
  27. [27] Mabuchi T., Mukai S., Stability and Einstein-Kähler metric of a quartic del Pezzo surface, In: Einstein Metrics and Yang-Mills Connections, Sanda, December 6–11, 1990, Lecture Notes in Pure and Appl. Math., 145, Dekker, New York, 1993, 133–160 Zbl0809.53070
  28. [28] Markushevich D., Tikhomirov A.S., The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., 2001, 10(1), 37–62 Zbl0987.14028
  29. [29] Mukai S., Curves and symmetric spaces, I, Amer. J. Math., 1995, 117(6), 1627–1644 http://dx.doi.org/10.2307/2375032 Zbl0871.14025
  30. [30] Namikawa Y., Smoothing Fano 3-folds, J. Algebraic Geom., 1997, 6(2), 307–324 Zbl0906.14019
  31. [31] Reid M., The complete intersection of two or more quadrics, PhD thesis, Cambridge University, 1972, availble at http://homepages.warwick.ac.uk/∼masda/3folds/qu.pdf 
  32. [32] Shatz S.S., On subbundles of vector bundles over P 1, J. Pure Appl. Algebra, 1977/78, 10(3), 315–322 http://dx.doi.org/10.1016/0022-4049(77)90010-X 
  33. [33] Shramov K.A., On the rationality of nonsingular threefolds with a pencil of del Pezzo surfaces of degree 4, Sb. Math., 2006, 197(1–2), 127–137 http://dx.doi.org/10.1070/SM2006v197n01ABEH003749 Zbl1134.14310
  34. [34] Tjurin A.N., The intersection of quadrics, Uspehi Mat. Nauk, 1975, 30(6), 51–99 
  35. [35] Zhu Y., Homogeneous fibrations over curves, preprint available at http://arxiv.org/abs/1111.2963 

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.