Rational curves on prime Fano threefolds of index 1
Authors:
Brian Lehmann and Sho Tanimoto
Journal:
J. Algebraic Geom. 30 (2021), 151-188
DOI:
https://doi.org/10.1090/jag/751
Published electronically:
December 9, 2019
MathSciNet review:
4233180
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Abstract |
References |
Additional Information
Abstract: We study the moduli spaces of rational curves on prime Fano threefolds of index 1. For general threefolds of most genera we compute the dimension and the number of irreducible components of these moduli spaces. Our results confirm Geometric Manin’s Conjecture in these examples and show the enumerativity of certain Gromov-Witten invariants.
References
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- Aaron Bertram and Holger P. Kley, New recursions for genus-zero Gromov-Witten invariants, Topology 44 (2005), no. 1, 1–24. MR 2103998, DOI https://doi.org/10.1016/j.top.2001.12.002
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- Izzet Coskun and Eric Riedl, Normal bundles of rational curves on complete intersections, Commun. Contemp. Math. 21 (2019), no. 2, 1850011, 29. MR 3918047, DOI https://doi.org/10.1142/S0219199718500116
- Olivier Debarre, Atanas Iliev, and Laurent Manivel, On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom. 21 (2012), no. 1, 21–59. MR 2846678, DOI https://doi.org/10.1090/S1056-3911-2011-00594-8
- Andreas Gathmann, Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), no. 2, 171–203. MR 1944571, DOI https://doi.org/10.1215/S0012-7094-02-11521-X
- Vasily V. Golyshev, Classification problems and mirror duality, Surveys in geometry and number theory: reports on contemporary Russian mathematics, London Math. Soc. Lecture Note Ser., vol. 338, Cambridge Univ. Press, Cambridge, 2007, pp. 88–121. MR 2306141, DOI https://doi.org/10.1017/CBO9780511721472.004
- Christopher D. Hacon and Chen Jiang, On Fujita invariants of subvarieties of a uniruled variety, Algebr. Geom. 4 (2017), no. 3, 304–310. MR 3652082, DOI https://doi.org/10.14231/AG-2017-017
- Joe Harris, Mike Roth, and Jason Starr, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106. MR 2070144, DOI https://doi.org/10.1515/crll.2004.045
- Brendan Hassett, Sho Tanimoto, and Yuri Tschinkel, Balanced line bundles and equivariant compactifications of homogeneous spaces, Int. Math. Res. Not. IMRN 15 (2015), 6375–6410. MR 3384482, DOI https://doi.org/10.1093/imrn/rnu129
- Atanas Iliev and Laurent Manivel, Prime Fano threefolds and integrable systems, Math. Ann. 339 (2007), no. 4, 937–955. MR 2341908, DOI https://doi.org/10.1007/s00208-007-0145-8
- V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
- Seán Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR 1610249, DOI https://doi.org/10.1090/memo/0669
- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math. 13 (2018), no. 1, 109–185. MR 3776469, DOI https://doi.org/10.1007/s11537-017-1714-6
- János Kollár and Frank-Olaf Schreyer, Real Fano 3-folds of type $V_{22}$, The Fano Conference, Univ. Torino, Turin, 2004, pp. 515–531. MR 2112589
- A. G. Kuznetsov, Derived categories of the Fano threefolds $V_{12}$, Mat. Zametki 78 (2005), no. 4, 579–594 (Russian, with Russian summary); English transl., Math. Notes 78 (2005), no. 3-4, 537–550. MR 2226730, DOI https://doi.org/10.1007/s11006-005-0152-6
- M. Letizia, The Abel-Jacobi mapping for the quartic threefold, Invent. Math. 75 (1984), no. 3, 477–492. MR 735337, DOI https://doi.org/10.1007/BF01388640
- B. Lehmann, A.K. Sengupta, and S. Tanimoto. Geometric consistency of Manin’s Conjecture. arXiv:1805.10580 [math.AG], 2018.
- Brian Lehmann and Sho Tanimoto, On the geometry of thin exceptional sets in Manin’s conjecture, Duke Math. J. 166 (2017), no. 15, 2815–2869. MR 3712166, DOI https://doi.org/10.1215/00127094-2017-0011
- Brian Lehmann and Sho Tanimoto, Geometric Manin’s conjecture and rational curves, Compos. Math. 155 (2019), no. 5, 833–862. MR 3937701, DOI https://doi.org/10.1112/s0010437x19007103
- Brian Lehmann, Sho Tanimoto, and Yuri Tschinkel, Balanced line bundles on Fano varieties, J. Reine Angew. Math. 743 (2018), 91–131. MR 3859270, DOI https://doi.org/10.1515/crelle-2015-0084
- B. G. Moĭšezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268 (Russian). MR 0213351
- V. V. Przhiyalkovskiĭ, Gromov-Witten invariants of Fano threefolds of genera 6 and 8, Mat. Sb. 198 (2007), no. 3, 145–158 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 3-4, 433–446. MR 2354283, DOI https://doi.org/10.1070/SM2007v198n03ABEH003843
- A. K. Sengupta, Manin’s conjecture and the Fujita invariant of finite covers, arXiv:1712.07780 [math.AG], 2017.
- Akash Kumar Sengupta, Manin’s $b$-constant in families, Algebra Number Theory 13 (2019), no. 8, 1893–1905. MR 4017538, DOI https://doi.org/10.2140/ant.2019.13.1893
- Mingmin Shen, On the normal bundles of rational curves on Fano 3-folds, Asian J. Math. 16 (2012), no. 2, 237–270. MR 2916363, DOI https://doi.org/10.4310/AJM.2012.v16.n2.a4
- Damiano Testa, The irreducibility of the spaces of rational curves on del Pezzo surfaces, J. Algebraic Geom. 18 (2009), no. 1, 37–61. MR 2448278, DOI https://doi.org/10.1090/S1056-3911-08-00484-0
References
- Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom. 22 (2013), no. 2, 201–248. MR 3019449, DOI https://doi.org/10.1090/S1056-3911-2012-00574-8
- Aaron Bertram and Holger P. Kley, New recursions for genus-zero Gromov-Witten invariants, Topology 44 (2005), no. 1, 1–24. MR 2103998, DOI https://doi.org/10.1016/j.top.2001.12.002
- V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann. 286 (1990), no. 1-3, 27–43 (French). MR 1032922, DOI https://doi.org/10.1007/BF01453564
- Ivan Cheltsov, Jihun Park, and Joonyeong Won, Cylinders in del Pezzo surfaces, Int. Math. Res. Not. IMRN 4 (2017), 1179–1230. MR 3658164, DOI https://doi.org/10.1093/imrn/rnw063
- Izzet Coskun and Eric Riedl, Normal bundles of rational curves on complete intersections, Commun. Contemp. Math. 21 (2019), no. 2, 1850011, 29. MR 3918047, DOI https://doi.org/10.1142/S0219199718500116
- Olivier Debarre, Atanas Iliev, and Laurent Manivel, On the period map for prime Fano threefolds of degree 10, J. Algebraic Geom. 21 (2012), no. 1, 21–59. MR 2846678, DOI https://doi.org/10.1090/S1056-3911-2011-00594-8
- Andreas Gathmann, Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), no. 2, 171–203. MR 1944571, DOI https://doi.org/10.1215/S0012-7094-02-11521-X
- Vasily V. Golyshev, Classification problems and mirror duality, Surveys in geometry and number theory: reports on contemporary Russian mathematics, London Math. Soc. Lecture Note Ser., vol. 338, Cambridge Univ. Press, Cambridge, 2007, pp. 88–121. MR 2306141, DOI https://doi.org/10.1017/CBO9780511721472.004
- Christopher D. Hacon and Chen Jiang, On Fujita invariants of subvarieties of a uniruled variety, Algebr. Geom. 4 (2017), no. 3, 304–310. MR 3652082, DOI https://doi.org/10.14231/AG-2017-017
- Joe Harris, Mike Roth, and Jason Starr, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106. MR 2070144, DOI https://doi.org/10.1515/crll.2004.045
- Brendan Hassett, Sho Tanimoto, and Yuri Tschinkel, Balanced line bundles and equivariant compactifications of homogeneous spaces, Int. Math. Res. Not. IMRN 15 (2015), 6375–6410. MR 3384482, DOI https://doi.org/10.1093/imrn/rnu129
- Atanas Iliev and Laurent Manivel, Prime Fano threefolds and integrable systems, Math. Ann. 339 (2007), no. 4, 937–955. MR 2341908, DOI https://doi.org/10.1007/s00208-007-0145-8
- V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
- Seán Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR 1610249, DOI https://doi.org/10.1090/memo/0669
- János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180
- Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov, Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math. 13 (2018), no. 1, 109–185. MR 3776469, DOI https://doi.org/10.1007/s11537-017-1714-6
- János Kollár and Frank-Olaf Schreyer, Real Fano 3-folds of type $V_{22}$, The Fano Conference, Univ. Torino, Turin, 2004, pp. 515–531. MR 2112589
- A. G. Kuznetsov, Derived categories of the Fano threefolds $V_{12}$, Mat. Zametki 78 (2005), no. 4, 579–594 (Russian, with Russian summary); English transl., Math. Notes 78 (2005), no. 3-4, 537–550. MR 2226730, DOI https://doi.org/10.1007/s11006-005-0152-6
- M. Letizia, The Abel-Jacobi mapping for the quartic threefold, Invent. Math. 75 (1984), no. 3, 477–492. MR 735337, DOI https://doi.org/10.1007/BF01388640
- B. Lehmann, A.K. Sengupta, and S. Tanimoto. Geometric consistency of Manin’s Conjecture. arXiv:1805.10580 [math.AG], 2018.
- Brian Lehmann and Sho Tanimoto, On the geometry of thin exceptional sets in Manin’s conjecture, Duke Math. J. 166 (2017), no. 15, 2815–2869. MR 3712166, DOI https://doi.org/10.1215/00127094-2017-0011
- Brian Lehmann and Sho Tanimoto, Geometric Manin’s conjecture and rational curves, Compos. Math. 155 (2019), no. 5, 833–862. MR 3937701, DOI https://doi.org/10.1112/s0010437x19007103
- Brian Lehmann, Sho Tanimoto, and Yuri Tschinkel, Balanced line bundles on Fano varieties, J. Reine Angew. Math. 743 (2018), 91–131. MR 3859270, DOI https://doi.org/10.1515/crelle-2015-0084
- B. G. Moĭšezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 225–268 (Russian). MR 0213351
- V. V. Przhiyalkovskiĭ, Gromov-Witten invariants of Fano threefolds of genera 6 and 8, Mat. Sb. 198 (2007), no. 3, 145–158 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 3-4, 433–446. MR 2354283, DOI https://doi.org/10.1070/SM2007v198n03ABEH003843
- A. K. Sengupta, Manin’s conjecture and the Fujita invariant of finite covers, arXiv:1712.07780 [math.AG], 2017.
- Akash Kumar Sengupta, Manin’s b-constant in families, Algebra Number Theory 13 (2019), no. 8, 1893–1905. MR 4017538, DOI https://doi.org/10.2140/ant.2019.13.1893
- Mingmin Shen, On the normal bundles of rational curves on Fano 3-folds, Asian J. Math. 16 (2012), no. 2, 237–270. MR 2916363, DOI https://doi.org/10.4310/AJM.2012.v16.n2.a4
- Damiano Testa, The irreducibility of the spaces of rational curves on del Pezzo surfaces, J. Algebraic Geom. 18 (2009), no. 1, 37–61. MR 2448278, DOI https://doi.org/10.1090/S1056-3911-08-00484-0
Additional Information
Brian Lehmann
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
MR Author ID:
977848
Email:
lehmannb@bc.edu
Sho Tanimoto
Affiliation:
Department of Mathematics, Faculty of Science, Kumamoto University, Kurokami 2-39-1, Kumamoto 860-8555, Japan
MR Author ID:
973697
Email:
stanimoto@kumamoto-u.ac.jp
Received by editor(s):
August 15, 2018
Received by editor(s) in revised form:
September 6, 2018, and July 16, 2019
Published electronically:
December 9, 2019
Additional Notes:
The first author was supported by NSF grant 1600875. The second author was partially supported by Lars Hesselholt’s Niels Bohr professorship and by MEXT Japan, Leading Initiative for Excellent Young Researchers (LEADER), Inamori Foundation, and JSPS KAKENHI Early-Career Scientists Grant number 19K14512.
Article copyright:
© Copyright 2019
University Press, Inc.
