The probability that a complete intersection is smooth

Alina Bucur[1]; Kiran S. Kedlaya[1]

  • [1] University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 541-556
  • ISSN: 1246-7405

Abstract

top
Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.

How to cite

top

Bucur, Alina, and Kedlaya, Kiran S.. "The probability that a complete intersection is smooth." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 541-556. <http://eudml.org/doc/251045>.

@article{Bucur2012,
abstract = {Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.},
affiliation = {University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093; University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093},
author = {Bucur, Alina, Kedlaya, Kiran S.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Bertini theorem; smooth subscheme of a projective space; finite field; hypersurface of large degree; complete intersection; random smooth intersection; probabilistic model},
language = {eng},
month = {11},
number = {3},
pages = {541-556},
publisher = {Société Arithmétique de Bordeaux},
title = {The probability that a complete intersection is smooth},
url = {http://eudml.org/doc/251045},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Bucur, Alina
AU - Kedlaya, Kiran S.
TI - The probability that a complete intersection is smooth
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 541
EP - 556
AB - Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.
LA - eng
KW - Bertini theorem; smooth subscheme of a projective space; finite field; hypersurface of large degree; complete intersection; random smooth intersection; probabilistic model
UR - http://eudml.org/doc/251045
ER -

References

top
  1. P. Billingsley, Probability and Measure, second edition. Wiley, New York, 1986. Zbl0411.60001MR830424
  2. B.W. Brock and A. Granville, More points than expected on curves over finite field extensions. Finite Fields Appl. 7 (2001), no. 1, 70–91. Zbl1023.11029MR1803936
  3. A. Bucur, C. David, B. Feigon, and M. Lalín, Statistics for traces of cyclic trigonal curves over finite fields. Int. Math. Res. Not. No. 5 (2010), 932–967. Zbl1201.11063MR2595014
  4. A. Bucur, C. David, B. Feigon, and M. Lalín, The fluctuations in the number of points of smooth plane curves over finite fields. J. Number Theory 130 (2010), 2528–2541. Zbl1210.11070MR2678860
  5. O. Gabber, On space filling curves and Albanese varieties. Geom. Funct. Anal. 11 (2001), 1192–1200. Zbl1072.14513MR1878318
  6. A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices No. 19 (1998), 991–1009. Zbl0924.11018MR1654759
  7. 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. Zbl1221.11141MR2488590
  8. P. Kurlberg and I. Wigman, Gaussian point count statistics for families of curves over a fixed finite field. Int. Math. Res. Not. 2011 (2011), 2217–2229. Zbl1297.11063MR2806563
  9. S. Lang and A. Weil, Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827. Zbl0058.27202MR65218
  10. B. Poonen, Squarefree values of multivariable polynomials. Duke Math. J. 118 (2003), 353–373. Zbl1047.11021MR1980998
  11. B. Poonen, Bertini theorems over finite fields. Ann. Math. 160 (2004), 1099–1127. Zbl1084.14026MR2144974
  12. M.M. Wood, The distribution of the number of points on trigonal curves over 𝔽 q . Int. Math. Res. Not. IMRN 2012. To appear, doi: 10.1093/imrn/rnr256. 

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.