Geometry & Topology Volume 26, issue 6 (2022)

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Classifying sections of del Pezzo fibrations, II

Brian Lehmann and Sho Tanimoto

Geometry & Topology 26 (2022) 2565–2647

DOI: 10.2140/gt.2022.26.2565

Bibliography

1	I Arzhantsev, U Derenthal, J Hausen, A Laface, Cox rings, 144, Cambridge Univ. Press (2015) MR3307753
2	E Ballico, On the Hilbert scheme of curves in a smooth quadric, from: "Deformations of mathematical structures" (editor J Ławrynowicz), Kluwer (1989) 127 MR987730
3	V Batyrev, Distribution of rational points of bounded height, lecture (1988)
4	V V Batyrev, Y I Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann. 286 (1990) 27 MR1032922
5	R Beheshti, B Lehmann, E Riedl, S Tanimoto, Moduli spaces of rational curves on Fano threefolds, Adv. Math. 408 (2022) MR4456787
6	S Bloch, V Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math. 105 (1983) 1235 MR714776
7	D Bourqui, Produit eulérien motivique et courbes rationnelles sur les variétés toriques, Compos. Math. 145 (2009) 1360 MR2575087
8	D Bourqui, Asymptotic behaviour of rational curves, preprint (2011) arXiv:1107.3824
9	D Bourqui, Algebraic points, non-anticanonical heights and the Severi problem on toric varieties, Proc. Lond. Math. Soc. 113 (2016) 474 MR3556489
10	T Browning, W Sawin, A geometric version of the circle method, Ann. of Math. 191 (2020) 893 MR4088353
11	T Browning, P Vishe, Rational curves on smooth hypersurfaces of low degree, Algebra Number Theory 11 (2017) 1657 MR3697151
12	A Bruguières, The scheme of morphisms from an elliptic curve to a Grassmannian, Compos. Math. 63 (1987) 15 MR906377
13	A M Castravet, Rational families of vector bundles on curves, Int. J. Math. 15 (2004) 13 MR2039210
14	A Corti, Del Pezzo surfaces over Dedekind schemes, Ann. of Math. 144 (1996) 641 MR1426888
15	I Coskun, J Starr, Rational curves on smooth cubic hypersurfaces, Int. Math. Res. Not. 2009 (2009) 4626 MR2564370
16	J S Ellenberg, A Venkatesh, Counting extensions of function fields with bounded discriminant and specified Galois group, from: "Geometric methods in algebra and number theory" (editors F Bogomolov, Y Tschinkel), Progr. Math. 235, Birkhäuser (2005) 151 MR2159381
17	T Graber, J Harris, J Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 57 MR1937199
18	J Harris, M Roth, J Starr, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004) 73 MR2070144
19	R Hartshorne, Algebraic geometry, 52, Springer (1977) MR0463157
20	B Hassett, S Tanimoto, Y Tschinkel, Balanced line bundles and equivariant compactifications of homogeneous spaces, Int. Math. Res. Not. 2015 (2015) 6375 MR3384482
21	F Hidaka, K Watanabe, Normal Gorenstein surfaces with ample anti-canonical divisor, Tokyo J. Math. 4 (1981) 319 MR646042
22	D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, E31, Vieweg & Sohn (1997) MR1450870
23	S Ishii, Moduli space of polarized del Pezzo surfaces and its compactification, Tokyo J. Math. 5 (1982) 289 MR688127
24	Y Kawamata, Crepant blowing-up of 3–dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988) 93 MR924674
25	B Kim, R Pandharipande, The connectedness of the moduli space of maps to homogeneous spaces, from: "Symplectic geometry and mirror symmetry" (editors K Fukaya, Y G Oh, K Ono, G Tian), World Sci. (2001) 187 MR1882330
26	J Kollár, Rational curves on algebraic varieties, 32, Springer (1996) MR1440180
27	J Kollár, S Mori, Birational geometry of algebraic varieties, 134, Cambridge Univ. Press (1998) MR1658959
28	B Lehmann, A K Sengupta, S Tanimoto, Geometric consistency of Manin’s conjecture, Compos. Math. 158 (2022) 1375 MR4472281
29	B Lehmann, S Tanimoto, On the geometry of thin exceptional sets in Manin’s conjecture, Duke Math. J. 166 (2017) 2815 MR3712166
30	B Lehmann, S Tanimoto, Classifying sections of del Pezzo fibrations, I, preprint (2019) arXiv:1912.04369
31	B Lehmann, S Tanimoto, Geometric Manin’s conjecture and rational curves, Compos. Math. 155 (2019) 833 MR3937701
32	B Lehmann, S Tanimoto, On exceptional sets in Manin’s conjecture, Res. Math. Sci. 6 (2019) MR3893027
33	B Lehmann, S Tanimoto, Rational curves on prime Fano threefolds of index 1, J. Algebraic Geom. 30 (2021) 151 MR4233180
34	Y I Manin, Problems on rational points and rational curves on algebraic varieties, from: "Surveys in differential geometry, II" (editors C C Hsiung, S T Yau), International (1995) 214 MR1375257
35	K Matsuki, Introduction to the Mori program, Springer (2002) MR1875410
36	Y Mustopa, M Teixidor i Bigas, Rational curves on moduli spaces of vector bundles, preprint (2020) arXiv:2007.10511
37	M Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math. J. 37 (1970) 191 MR258829
38	B Pasquier, N Perrin, Elliptic curves on some homogeneous spaces, Doc. Math. 18 (2013) 679 MR3084562
39	N Perrin, Elliptic curves on spinor varieties, Cent. Eur. J. Math. 10 (2012) 1393 MR2925611
40	E Riedl, D Yang, Kontsevich spaces of rational curves on Fano hypersurfaces, J. Reine Angew. Math. 748 (2019) 207 MR3918434
41	M Shen, On the normal bundles of rational curves on Fano 3–folds, Asian J. Math. 16 (2012) 237 MR2916363
42	D Testa, The irreducibility of the spaces of rational curves on del Pezzo surfaces, J. Algebraic Geom. 18 (2009) 37 MR2448278
43	J F Thomsen, Irreducibility of M_0,n(G∕P,β), Int. J. Math. 9 (1998) 367 MR1625369
44	Y Tschinkel, Algebraic varieties with many rational points, from: "Arithmetic geometry" (editors H Darmon, D A Ellwood, B Hassett, Y Tschinkel), Clay Math. Proc. 8, Amer. Math. Soc. (2009) 243 MR2498064
45	C Voisin, Hodge theory and complex algebraic geometry, II, 77, Cambridge Univ. Press (2003) MR1997577
46	C Voisin, On integral Hodge classes on uniruled or Calabi–Yau threefolds, from: "Moduli spaces and arithmetic geometry" (editors S Mukai, Y Miyaoka, S Mori, A Moriwaki, I Nakamura), Adv. Stud. Pure Math. 45, Math. Soc. Japan (2006) 43 MR2306166