Algebra & Number Theory

Rational curves on smooth hypersurfaces of low degree

Timothy Browning and Pankaj Vishe

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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 establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.

Article information

Algebra Number Theory, Volume 11, Number 7 (2017), 1657-1675.

Received: 2 November 2016
Revised: 27 March 2017
Accepted: 23 May 2017
First available in Project Euclid: 12 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic)
Secondary: 11P55: Applications of the Hardy-Littlewood method [See also 11D85] 14G05: Rational points

rational curves circle method function fields hypersurfaces


Browning, Timothy; Vishe, Pankaj. Rational curves on smooth hypersurfaces of low degree. Algebra Number Theory 11 (2017), no. 7, 1657--1675. doi:10.2140/ant.2017.11.1657.

Export citation


  • R. Beheshti, “Hypersurfaces with too many rational curves”, Math. Ann. 360:3–4 (2014), 753–768.
  • R. Beheshti and N. M. Kumar, “Spaces of rational curves on complete intersections”, Compos. Math. 149:6 (2013), 1041–1060.
  • D. Bourqui, “Moduli spaces of curves and Cox rings”, Michigan Math. J. 61:3 (2012), 593–613.
  • D. Bourqui, “Exemples de comptages de courbes sur les surfaces”, Math. Ann. 357:4 (2013), 1291–1327.
  • T. D. Browning and D. R. Heath-Brown, “Rational points on quartic hypersurfaces”, J. Reine Angew. Math. 629 (2009), 37–88.
  • T. D. Browning and P. Vishe, “Rational points on cubic hypersurfaces over $\mathbb{F}_q(t)$”, Geom. Funct. Anal. 25:3 (2015), 671–732.
  • I. Coskun and J. Starr, “Rational curves on smooth cubic hypersurfaces”, Int. Math. Res. Not. 2009:24 (2009), 4626–4641.
  • A. Grothendieck, “Eléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, III”, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255.
  • J. Harris, M. Roth, and J. Starr, “Rational curves on hypersurfaces of low degree”, J. Reine Angew. Math. 571 (2004), 73–106.
  • J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik $($3$)$ 32, Springer, Berlin, 1996.
  • S. Lang and A. Weil, “Number of points of varieties in finite fields”, Amer. J. Math. 76 (1954), 819–827.
  • S.-l. A. Lee, “Birch's theorem in function fields”, preprint, 2011.
  • S.-l. A. Lee, On the applications of the circle method to function fields, and related topics, Ph.D. thesis, University of Bristol, 2013,
  • T. Pugin, An algebraic circle method, Ph.D. thesis, Columbia University, 2011,
  • E. Riedl and Y. Yang, “Kontsevich spaces of rational curves on Fano hypersurfaces”, J. Reine Angew. Math. (online publication August 2016).
  • J.-P. Serre, “How to use finite fields for problems concerning infinite fields”, pp. 183–193 in Arithmetic, geometry, cryptography and coding theory (Marseilles, 2007), edited by G. Lachaud et al., Contemp. Math. 487, Amer. Math. Soc., Providence, RI, 2009.
  • M. Usher and J. Zhang, “Persistent homology and Floer–Novikov theory”, Geom. Topol. 20:6 (2016), 3333–3430.