Geometric Manin’s conjecture and rational curves. (English) Zbl 1451.14089
The article proves several interesting results about family of rational curves on a Fano variety. These have certain expected dimension and it is interesting to know whether it is the real dimension, and often it is not, what can be said about the ‘bad’ components and what are they. To state one of the theorems, one needs to define the Fujita invariant \(a(X,L)\) for a projective variety \(X\) and a \(\mathbb{Q}\)-Cartier nef divisor \(L\). This is the minimum real number \(t\) such that \(tL+K_X\) is in the cone of pseudo-effective divisors. Then the theorem says that the union of all \(Y\subset X\) (\(X\) Fano with \(L=-K_X\)) with \(a(Y,L_{|Y})>a(X,L)\) is a proper closed subvariety and the rational curves of a fixed degree with respect to \(L\) and not contained in this closed subvariety has the expected dimension.
Reviewer: N. Mohan Kumar (St. Louis)
References:
| [1] | Andreatta, M., Minimal model program with scaling and adjunction theory, Internat. J. Math., 24, (2013) · Zbl 1273.14030 |
| [2] | 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), 201-248. · Zbl 1267.32017 |
| [3] | Birch, B. J., Forms in many variables, Proc. Roy. Soc. Edinburgh Sect. A, 265, 245-263, (19611962) · Zbl 0103.03102 |
| [4] | Birkar, C., Singularities of linear systems and boundedness of Fano varieties, Preprint (2016), arXiv:1609.05543 [math.AG]. |
| [5] | Beheshti, R. and Mohan Kumar, N., Spaces of rational curves on complete intersections, Compos. Math.149 (2013), 1041-1060. · Zbl 1278.14067 |
| [6] | Browning, T. D. and Loughran, D., Varieties with too many rational points, Math. Z.285 (2017), 1249-1267. · Zbl 1457.11094 |
| [7] | Batyrev, V. V. and Manin, Yu. I., Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann.286 (1990), 27-43. · Zbl 0679.14008 |
| [8] | Behrend, K. and Manin, Yu., Stacks of stable maps and Gromov-Witten invariants, Duke Math. J.85 (1996), 1-60. · Zbl 0872.14019 |
| [9] | Bourqui, D., Asymptotic behaviour of rational curves, Preprint (2011), arXiv:1107.3824. |
| [10] | Bourqui, D., Moduli spaces of curves and Cox rings, Michigan Math. J., 61, 593-613, (2012) · Zbl 1258.14014 |
| [11] | Bourqui, D., Exemples de comptages de courbes sur les surfaces, Math. Ann., 357, 1291-1327, (2013) · Zbl 1361.14025 |
| [12] | Bourqui, D., Algebraic points, non-anticanonical heights and the Severi problem on toric varieties, Proc. Lond. Math. Soc. (3), 113, 474-514, (2016) · Zbl 1427.14108 |
| [13] | Batyrev, V. V. and Tschinkel, Y., Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris Sér. I Math.323 (1996), 41-46. · Zbl 0879.14007 |
| [14] | Browning, T. and Vishe, P., Rational curves on hypersurfaces of low degree, Algebra Number Theory11 (2017), 1657-1675. · Zbl 1442.14094 |
| [15] | Campana, F., Connexité rationnelle des variétés de Fano, Ann. Sci. Éc. Norm. Supér. (4), 25, 539-545, (1992) · Zbl 0783.14022 |
| [16] | Castravet, A.-M., Rational families of vector bundles on curves, Internat. J. Math., 15, 13-45, (2004) · Zbl 1092.14041 |
| [17] | Cheltsov, I., Park, J. and Won, J., Cylinders in del Pezzo surfaces, Int. Math. Res. Not. IMRN2017 (2017), 1179-1230. · Zbl 1405.14089 |
| [18] | Coskun, I. and Starr, J., Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. IMRN2009 (2009), 4626-4641. · Zbl 1200.14051 |
| [19] | 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, (Birkhäuser, Boston, 2005), 151-168. · Zbl 1085.11057 |
| [20] | Fujita, T., Remarks on quasi-polarized varieties, Nagoya Math. J., 115, 105-123, (1989) · Zbl 0699.14002 |
| [21] | 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), 249-276. |
| [22] | Guerra, L., Complexity of Chow varieties and number of morphisms on surfaces of general type, Manuscripta Math., 98, 1-8, (1999) · Zbl 0977.14001 |
| [23] | Höring, A., The sectional genus of quasi-polarised varieties, Arch. Math. (Basel), 95, 125-133, (2010) · Zbl 1198.14008 |
| [24] | Hacon, C. and Jiang, C., On Fujita invariants of subvarieties of a uniruled variety, Algebr. Geom.4 (2017), 304-310. · Zbl 1370.14007 |
| [25] | Harris, J., Roth, M. and Starr, J., Rational curves on hypersurfaces of low degree, J. Reine Angew. Math.571 (2004), 73-106. · Zbl 1052.14027 |
| [26] | Hassett, B., Tanimoto, S. and Tschinkel, Y., Balanced line bundles and equivariant compactifications of homogeneous spaces, Int. Math. Res. Not. IMRN2015 (2015), 6375-6410. · Zbl 1354.14075 |
| [27] | 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, 703-712, (2005) · Zbl 1083.14062 |
| [28] | Iskovskih, V. A., Anticanonical models of three-dimensional algebraic varieties, in Current problems in mathematics, Vol. 12 (Russian) (VINITI, Moscow, 1979), 59-157; 239 (loose errata). · Zbl 0415.14024 |
| [29] | Iskovskikh, V. A. and Prokhorov, Yu. G., Fano varieties, in Algebraic geometry, V, (Springer, Berlin, 1999), 1-247. · Zbl 0912.14013 |
| [30] | Keel, S. and Mckernan, J., Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc.140 (1999). · Zbl 0955.14031 |
| [31] | Kollár, J., Miyaoka, Y. and Mori, S., Rational connectedness and boundedness of Fano manifolds, J. Differential Geom.36 (1992), 765-779. · Zbl 0759.14032 |
| [32] | Kollár, J., Rational curves on algebraic varieties, (1996), Springer: Springer, Berlin |
| [33] | Kollár, J., The Lefschetz property for families of curves, in Rational points, rational curves, and entire holomorphic curves on projective varieties, (American Mathematical Society, Providence, RI, 2015), 143-154. · Zbl 1470.14052 |
| [34] | 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), 187-201. · Zbl 1076.14517 |
| [35] | Lehmann, B. and Tanimoto, S., On the geometry of thin exceptional sets in Manin’s conjecture, Duke Math. J.166 (2017), 2815-2869. · Zbl 1451.14072 |
| [36] | Lehmann, B. and Tanimoto, S., On exceptional sets in Manin’s Conjecture, Res. Math. Sci.6(1) (2019), paper No. 12. · Zbl 1431.14019 |
| [37] | Lehmann, B., Tanimoto, S. and Tschinkel, Y., Balanced line bundles on Fano varieties, J. Reine Angew. Math.743 (2018), 91-131. · Zbl 1397.14029 |
| [38] | 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), 214-245. · Zbl 0847.14012 |
| [39] | 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). · Zbl 0568.14023 |
| [40] | Miyanishi, M. and Zhang, D.-Q., Gorenstein log del Pezzo surfaces of rank one, J. Algebra118 (1988), 63-84. · Zbl 0664.14019 |
| [41] | Peyre, E., Points de hauteur bornée, topologie adélique et mesures de Tamagawa, J. Théor. Nombres Bordeaux, 15, 319-349, (2003) · Zbl 1057.14031 |
| [42] | Rudulier, C. L., Points algébriques de hauteur bornée sur une surface, 2014,http://cecile.lerudulier.fr/Articles/surfaces.pdf. · Zbl 1338.14028 |
| [43] | Riedl, E. and Yang, D., Kontsevich spaces of rational curves on Fano hypersurfaces, J. Reine Agnew. Math.748 (2019), 207-225. · Zbl 1410.14026 |
| [44] | Sanna, G., Rational curves and instantons on the Fano threefold\(Y_5 \). PhD thesis, Scuola Internazionale di Studi Superiori Avanzati (2013/2014). |
| [45] | 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. |
| [46] | Testa, D., The irreducibility of the spaces of rational curves on del Pezzo surfaces, J. Algebraic Geom., 18, 37-61, (2009) · Zbl 1165.14024 |
| [47] | Thomsen, J. F., Irreducibility of \(\overline{\mbox{ M }}_{0, n \) · Zbl 0934.14020 |
| [48] | Tschinkel, Y., Algebraic varieties with many rational points, in Arithmetic geometry, (American Mathematical Society, Providence, RI, 2009), 243-334. · Zbl 1193.14030 |
| [49] | Tian, Z. and Zong, H. R., One-cycles on rationally connected varieties, Compos. Math.150 (2014), 396-408. · Zbl 1304.14067 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
