On arithmetic intersection numbers on self-products of curves
Author:
Robert Wilms
Journal:
J. Algebraic Geom. 31 (2022), 397-424
DOI:
https://doi.org/10.1090/jag/777
Published electronically:
January 4, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number $\hat {\omega }^2$ of the dualizing sheaf of a curve in terms of Zhang’s invariant $\varphi$. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.
References
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References
- J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), no. 4, 903–1027. MR 1260106, DOI 10.2307/2152736
- Zubeyir Cinkir, Zhang’s conjecture and the effective Bogomolov conjecture over function fields, Invent. Math. 183 (2011), no. 3, 517–562. MR 2772087, DOI 10.1007/s00222-010-0282-7
- Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174 (1991). MR 1087394
- H. Gillet and C. Soulé, Arithmetic analogs of the standard conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 129–140. MR 1265527, DOI 10.1090/pspum/055.1/1265527
- Walter Gubler, Höhentheorie, Math. Ann. 298 (1994), no. 3, 427–455 (German). With an appendix by Jürg Kramer. MR 1262769, DOI 10.1007/BF01459743
- Joe Harris, Curves and their moduli, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 99–143. MR 927953
- Robin de Jong, Néron-Tate heights of cycles on Jacobians, J. Algebraic Geom. 27 (2018), no. 2, 339–381. MR 3764279, DOI 10.1090/jag/700
- Laurent Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985), 266 (French, with English summary). MR 797982
- Patrice Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255–283 (French). MR 1092175, DOI 10.1007/BF01446571
- Emmanuel Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), no. 1, 167–179 (French). MR 1609514, DOI 10.2307/120987
- Robert Wilms, New explicit formulas for Faltings’ delta-invariant, Invent. Math. 209 (2017), no. 2, 481–539. MR 3674221, DOI 10.1007/s00222-016-0713-1
- Xinyi Yuan and Shou-Wu Zhang, The arithmetic Hodge index theorem for adelic line bundles, Math. Ann. 367 (2017), no. 3-4, 1123–1171. MR 3623221, DOI 10.1007/s00208-016-1414-1
- Shouwu Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), no. 1, 171–193. MR 1207481, DOI 10.1007/BF01232429
- Shouwu Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), no. 2, 281–300. MR 1311351
- Shouwu Zhang, Heights and reductions of semi-stable varieties, Compositio Math. 104 (1996), no. 1, 77–105. MR 1420712
- Shou-Wu Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159–165. MR 1609518, DOI 10.2307/120986
- Shou-Wu Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2010), no. 1, 1–73. MR 2563759, DOI 10.1007/s00222-009-0209-3
Additional Information
Robert Wilms
Affiliation:
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
MR Author ID:
975037
Email:
robert.wilms@unibas.ch
Received by editor(s):
February 20, 2020
Received by editor(s) in revised form:
August 27, 2020
Published electronically:
January 4, 2022
Additional Notes:
The author was supported by SFB/Transregio 45.
Article copyright:
© Copyright 2022
University Press, Inc.
