Skip to main content Accessibility help
×
×
Home

Geometric Manin’s conjecture and rational curves

  • Brian Lehmann (a1) and Sho Tanimoto (a2) (a3)
Abstract

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli space of rational curves on $X$ using the perspective of Manin’s conjecture. In particular, we bound the dimension and number of components of spaces of rational curves on $X$ . We propose a geometric Manin’s conjecture predicting the growth rate of a counting function associated to the irreducible components of these moduli spaces.

Copyright
Footnotes
Hide All

Lehmann is supported by NSF grant 1600875. Tanimoto is partially supported by Lars Hesselholt’s Niels Bohr professorship, and MEXT Japan, Leading Initiative for Excellent Young Researchers (LEADER).

Footnotes
References
Hide All
[And13] Andreatta, M., Minimal model program with scaling and adjunction theory , Internat. J. Math. 24 (2013), 1350007.
[BDPP13] Boucksom, S., Demailly, J.-P., Paun, M. and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension , J. Algebraic Geom. 22 (2013), 201248.
[Bir62] Birch, B. J., Forms in many variables , Proc. Roy. Soc. Edinburgh Sect. A 265 (1961/1962), 245263.
[Bir16] Birkar, C., Singularities of linear systems and boundedness of Fano varieties, Preprint (2016), arXiv:1609.05543 [math.AG].
[BK13] Beheshti, R. and Mohan Kumar, N., Spaces of rational curves on complete intersections , Compos. Math. 149 (2013), 10411060.
[BL17] Browning, T. D. and Loughran, D., Varieties with too many rational points , Math. Z. 285 (2017), 12491267.
[BM90] Batyrev, V. V. and Manin, Yu. I., Sur le nombre des points rationnels de hauteur borné des variétés algébriques , Math. Ann. 286 (1990), 2743.
[BM96] Behrend, K. and Manin, Yu., Stacks of stable maps and Gromov-Witten invariants , Duke Math. J. 85 (1996), 160.
[Bou11] Bourqui, D., Asymptotic behaviour of rational curves, Preprint (2011), arXiv:1107.3824.
[Bou12] Bourqui, D., Moduli spaces of curves and Cox rings , Michigan Math. J. 61 (2012), 593613.
[Bou13] Bourqui, D., Exemples de comptages de courbes sur les surfaces , Math. Ann. 357 (2013), 12911327.
[Bou16] Bourqui, D., Algebraic points, non-anticanonical heights and the Severi problem on toric varieties , Proc. Lond. Math. Soc. (3) 113 (2016), 474514.
[BT96] Batyrev, V. V. and Tschinkel, Y., Rational points on some Fano cubic bundles , C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 4146.
[BV17] Browning, T. and Vishe, P., Rational curves on hypersurfaces of low degree , Algebra Number Theory 11 (2017), 16571675.
[Cam92] Campana, F., Connexité rationnelle des variétés de Fano , Ann. Sci. Éc. Norm. Supér. (4) 25 (1992), 539545.
[Cas04] Castravet, A.-M., Rational families of vector bundles on curves , Internat. J. Math. 15 (2004), 1345.
[CPW17] Cheltsov, I., Park, J. and Won, J., Cylinders in del Pezzo surfaces , Int. Math. Res. Not. IMRN 2017 (2017), 11791230.
[CS09] Coskun, I. and Starr, J., Rational curves on smooth cubic hypersurfaces , Int. Math. Res. Not. IMRN 2009 (2009), 46264641.
[EV05] Ellenberg, J. S. and Venkatesh, A., Counting extensions of function fields with bounded discriminant and specified Galois group , in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser, Boston, 2005), 151168.
[Fuj89] Fujita, T., Remarks on quasi-polarized varieties , Nagoya Math. J. 115 (1989), 105123.
[Gro95] Grothendieck, A., Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert , in Séminaire Bourbaki, Vol. 6, Exp. No. 221 (Société Mathématique de France, Paris, 1995), 249276.
[Gue99] Guerra, L., Complexity of Chow varieties and number of morphisms on surfaces of general type , Manuscripta Math. 98 (1999), 18.
[Hör10] Höring, A., The sectional genus of quasi-polarised varieties , Arch. Math. (Basel) 95 (2010), 125133.
[HJ17] Hacon, C. and Jiang, C., On Fujita invariants of subvarieties of a uniruled variety , Algebr. Geom. 4 (2017), 304310.
[HRS04] Harris, J., Roth, M. and Starr, J., Rational curves on hypersurfaces of low degree , J. Reine Angew. Math. 571 (2004), 73106.
[HTT15] Hassett, B., Tanimoto, S. and Tschinkel, Y., Balanced line bundles and equivariant compactifications of homogeneous spaces , Int. Math. Res. Not. IMRN 2015 (2015), 63756410.
[Hwa05] Hwang, J.-M., A bound on the number of curves of a given degree through a general point of a projective variety , Compos. Math. 141 (2005), 703712.
[Isk79] Iskovskih, V. A., Anticanonical models of three-dimensional algebraic varieties , in Current problems in mathematics, Vol. 12 (Russian) (VINITI, Moscow, 1979), 59157; 239 (loose errata).
[IP99] Iskovskikh, V. A. and Prokhorov, Yu. G., Fano varieties , in Algebraic geometry, V, Encyclopaedia of Mathematical Sciences, vol. 47 (Springer, Berlin, 1999), 1247.
[KM99] Keel, S. and McKernan, J., Rational curves on quasi-projective surfaces , Mem. Amer. Math. Soc. 140 (1999).
[KMM92] Kollár, J., Miyaoka, Y. and Mori, S., Rational connectedness and boundedness of Fano manifolds , J. Differential Geom. 36 (1992), 765779.
[Kol96] Kollár, J., Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 32 (Springer, Berlin, 1996).
[Kol15] Kollár, J., The Lefschetz property for families of curves , in Rational points, rational curves, and entire holomorphic curves on projective varieties, Contemporary Mathematics, vol. 654 (American Mathematical Society, Providence, RI, 2015), 143154.
[KP01] Kim, B. and Pandharipande, R., The connectedness of the moduli space of maps to homogeneous spaces , in Symplectic geometry and mirror symmetry (Seoul, 2000) (World Scientific, River Edge, NJ, 2001), 187201.
[LT17] Lehmann, B. and Tanimoto, S., On the geometry of thin exceptional sets in Manin’s conjecture , Duke Math. J. 166 (2017), 28152869.
[LT19] Lehmann, B. and Tanimoto, S., On exceptional sets in Manin’s Conjecture , Res. Math. Sci. 6(1) (2019), paper No. 12.
[LTT18] Lehmann, B., Tanimoto, S. and Tschinkel, Y., Balanced line bundles on Fano varieties , J. Reine Angew. Math. 743 (2018), 91131.
[Man95] Manin, Yu. I., Problems on rational points and rational curves on algebraic varieties , in Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) (International Press, Cambridge, MA, 1995), 214245.
[Mor84] Mori, S., Cone of curves, and Fano 3-folds , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) (PWN, Warsaw, 1984).
[MZ88] Miyanishi, M. and Zhang, D.-Q., Gorenstein log del Pezzo surfaces of rank one , J. Algebra 118 (1988), 6384.
[Pey03] Peyre, E., Points de hauteur bornée, topologie adélique et mesures de Tamagawa , J. Théor. Nombres Bordeaux 15 (2003), 319349.
[Rud14] Rudulier, C. L., Points algébriques de hauteur bornée sur une surface, 2014,http://cecile.lerudulier.fr/Articles/surfaces.pdf.
[RY19] Riedl, E. and Yang, D., Kontsevich spaces of rational curves on Fano hypersurfaces , J. Reine Agnew. Math. 748 (2019), 207225.
[San14] Sanna, G., Rational curves and instantons on the Fano threefold  $Y_{5}$ . PhD thesis, Scuola Internazionale di Studi Superiori Avanzati (2013/2014).
[Tes05] Testa, D., The Severi problem for rational curves on del Pezzo surfaces. PhD thesis, Massachusetts Institute of Technology (2005), https://arxiv.org/abs/math/0609355.
[Tes09] Testa, D., The irreducibility of the spaces of rational curves on del Pezzo surfaces , J. Algebraic Geom. 18 (2009), 3761.
[Tho98] Thomsen, J. F., Irreducibility of M 0, n (G/P, 𝛽) , Internat. J. Math. 9 (1998), 367376.
[Tsc09] Tschinkel, Y., Algebraic varieties with many rational points , in Arithmetic geometry, Clay Mathematics Proceedings, vol. 8 (American Mathematical Society, Providence, RI, 2009), 243334.
[TZ14] Tian, Z. and Zong, H. R., One-cycles on rationally connected varieties , Compos. Math. 150 (2014), 396408.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed