Abstract
In this chapter we review the basic definitions of Arakelov intersection theory, and then sketch the proofs of some fundamental results of Arakelov, Faltings and Hriljac. Many interesting topics are beyond the scope of this introduction, and may be found in the references [2], [3], [8], [12], [20] and their bibliographies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abhyankar, S. Resolutions of singularities of arithmetical surfaces, in Arithmetical Algebraic Geometry. Harper and Row: New York, 1965.
Arakelov, S. Intersection theory of divisors on an arithmetic surface. Izv. Akad. Nauk., 38 (1974), 1179–1192.
Arakelov, S. Theory of intersections on the arithmetic surface. Proceedings of the International Congress on Mathematics, Vancouver, 1974, pp. 405–408.
Artin, M. Lipman’s proof of resolution of singularities for surfaces, this volume, pp. 267–287.
Cantor, D. On an extension of the definition of transfinite diameter and some applications. J. Reine. Angew. Math., 316 (1980), 160–207.
Chinburg, T. Intersection theory and capacity theory on arithmetic surfaces. Proc. of the Canadian Math. Soc. Summer Seminar in Number Theory, 7. American Mathematical Society: Providence, RI, 1986.
Chinburg, T. Minimal models for curves over Dedekind rings, this volume, pp. 309–326.
Faltings, G. Calculus on arithmetic surfaces. Ann. Math., 119, no. 2 (1984), 387–424.
Hartshorne, R. Algebraic Geometry. Springer-Verlag: New York, 1977.
Hartshorne, R. Residues and Duality. Springer Lecture Notes, 20. Springer-Verlag: Heidelberg, 1966.
Hriljac, P. Thesis (1982), Massachusetts Institute of Technology.
Hriljac, P. Heights and Arakelov’s intersection theory. Amer. J. Math., 107, no. 1 (1985), 23–38.
Hriljac, P. The Arakelov adjunction formula. Columbia University, 1986. Preprint.
Lichtenbaum, S. Curves over discrete valuation rings. Amer. J. Math., 15, no. 2 (1968), 380–405.
Lipman, J. Rational singularities with applications to algebraic surfaces and unique factorization. Publ. Math. I.H.E.S., 36 (1969), 195–279.
Mumford, D. The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. I.H.E.S., 9 (1961), 5–22.
Raynaud, M. Specialization du functeur Picard. Publ. Math. I.H.E.S., 38 (1970), 27–76.
Rumely, R. Capacity Theory on Algebraic Curves. (To appear in Springer Lecture Notes in Mathematics.)
Silverman, J. The theory of height functions, this volume, pp. 151–166.
Szpiro, L. Degrée, intersections, hauteur, in Séminaire sur les Pinceaux Arithmetiques: La Conjecture de Mordell, L. Szpiro ed., Asterisque, 127, (1985), 11–28.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Chinburg, T. (1986). An Introduction to Arakelov Intersection Theory. In: Cornell, G., Silverman, J.H. (eds) Arithmetic Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8655-1_12
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8655-1_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8657-5
Online ISBN: 978-1-4613-8655-1
eBook Packages: Springer Book Archive