## Algebra & Number Theory

### Rational curves on smooth hypersurfaces of low degree

#### Abstract

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

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

Dates
Revised: 27 March 2017
Accepted: 23 May 2017
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.ant/1513096742

Digital Object Identifier
doi:10.2140/ant.2017.11.1657

Mathematical Reviews number (MathSciNet)
MR3697151

Zentralblatt MATH identifier
06775556

#### Citation

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. https://projecteuclid.org/euclid.ant/1513096742

#### References

• 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, http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.619268.
• T. Pugin, An algebraic circle method, Ph.D. thesis, Columbia University, 2011, https://search.proquest.com/docview/875798043.
• 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.