Duke Mathematical Journal

Semiample Bertini theorems over finite fields

Daniel Erman and Melanie Matchett Wood

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a semiample generalization of Poonen’s Bertini theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces and the distribution of points on those smooth curves.

Article information

Duke Math. J., Volume 164, Number 1 (2015), 1-38.

First available in Project Euclid: 9 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G15: Finite ground fields
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

Bertini theorems finite fields hyperplane sections


Erman, Daniel; Wood, Melanie Matchett. Semiample Bertini theorems over finite fields. Duke Math. J. 164 (2015), no. 1, 1--38. doi:10.1215/00127094-2838327. https://projecteuclid.org/euclid.dmj/1420813013

Export citation


  • [1] D. Bayer and D. Mumford, “What can be computed in algebraic geometry?” in Computational Algebraic Geometry and Commutative Algebra (Cortona, 1991), Sympos. Math. 34, Cambridge Univ. Press, Cambridge, 1993, 1–48.
  • [2] R. Becker and D. Glass, Pointless hyperelliptic curves, Finite Fields Appl. 21 (2013), 50–57.
  • [3] J. Bezerra, A. Garcia, and H. Stichtenoth, An explicit tower of function fields over cubic finite fields and Zink’s lower bound, J. Reine Angew. Math. 589 (2005), 159–199.
  • [4] A. Bucur, C. David, B. Feigon, and M. Lalín, Fluctuations in the number of points on smooth plane curves over finite fields, J. Number Theory 130 (2010), 2528–2541.
  • [5] A. Bucur, C. David, B. Feigon, and M. Lalín, Statistics for traces of cyclic trigonal curves over finite fields, Int. Math. Res. Not. IMRN 2010, no. 5, 932–967.
  • [6] A. Bucur, C. David, B. Feigon, and M. Lalín, “Biased statistics for traces of cyclic $p$-fold covers over finite fields” in WIN–Women in Numbers, Fields Inst. Commun. 60, Amer. Math. Soc., Providence, 2011, 121–143.
  • [7] A. Bucur and K. S. Kedlaya, The probability that a complete intersection is smooth, J. Théor. Nombres Bordeaux 24 (2012), 541–556.
  • [8] B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988), 116–138.
  • [9] D. Eisenbud, The Geometry of Syzygies, Grad. Texts in Math. 229, Springer, New York, 2005.
  • [10] N. D. Elkies, E. W. Howe, A. Kresch, B. Poonen, J. L. Wetherell, and M. E. Zieve, Curves of every genus with many points, II: Asymptotically good families, Duke Math. J. 122 (2004), 399–422.
  • [11] W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • [12] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [13] E. W. Howe, K. E. Lauter, and J. Top, “Pointless curves of genus three and four” in Arithmetic, Geometry and Coding Theory (AGCT 2003), Sémin. Congr. 11, Soc. Math. France, Paris, 2005, 125–141.
  • [14] J.-P. Jouanolou, Théorèmes de Bertini et applications, Progr. Math. 42, Birkhäuser, Boston, 1983.
  • [15] A. Kresch, J. L. Wetherell, and M. E. Zieve, Curves of every genus with many points, I: Abelian and toric families, J. Algebra 250 (2002), 353–370.
  • [16] P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field, J. Number Theory 129 (2009), 580–587.
  • [17] P. Kurlberg and I. Wigman, Gaussian point count statistics for families of curves over a fixed finite field, Int. Math. Res. Not. IMRN 2011, no. 10, 2217–2229.
  • [18] R. Lazarsfeld, Positivity in Algebraic Geometry, I, Ergeb. Math. Grenzgeb. (3) 48, Springer, Berlin, 2004.
  • [19] S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827.
  • [20] W.-C. W. Li and H. Maharaj, Coverings of curves with asymptotically many rational points, J. Number Theory 96 (2002), 232–256.
  • [21] L. Moret-Bailly, Bertini’s theorem in char p for base point free linear system, question answer, 24 August 2012, http://mathoverflow.net/questions/86163.
  • [22] N. H. Nguyen, Whitney theorems and Lefschetz pencils over finite fields, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2005.
  • [23] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), 1099–1127.
  • [24] B. Poonen, Sieve methods for varieties over finite fields and arithmetic schemes, J. Théor. Nombres Bordeaux 19 (2007), 221–229.
  • [25] B. Poonen, Smooth hypersurface sections containing a given subscheme over a finite field, Math. Res. Lett. 15 (2008), 265–271.
  • [26] H. M. Stark, “On the Riemann hypothesis in hyperelliptic function fields” in Analytic Number Theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, 1973, 285–302.
  • [27] H. Stichtenoth, Curves with a prescribed number of rational points, Finite Fields Appl. 17 (2011), 552–559.
  • [28] I. Swanson, Linear equivalence of ideal topologies, Math. Z. 234 (2000), 755–775.
  • [29] A. Temkine, Hilbert class field towers of function fields over finite fields and lower bounds for $A(q)$, J. Number Theory 87 (2001), 189–210.
  • [30] R. Vakil and M. M. Wood, Discriminants in the Grothendieck ring, to appear in Duke Math. J., preprint, arXiv:1208.3166v2 [math.AG].
  • [31] M. M. Wood. The distribution of the number of points on trigonal curves over $\mathfrak{f}_{q}$, Int. Math. Res. Not. IMRN 2012, no. 23, 5444–5456.
  • [32] Y. Zhao, On sieve methods for varieties over finite fields, Ph.D. dissertation, University of Wisconsin–Madison, Madison, Wis., 2013.