Abstract
The aim of this chapter is to give a new interpretation of the determinant method by means of Chow forms and techniques from Mumford’s geometric invariant theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
E. Bombieri, J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59, 337–357 (1989)
N. Broberg, A note on a paper by R. Heath-Brown: “the density of rational points on curves and surfaces”. J. Reine Angew. Math. 571, 159–178 (2004)
T.D. Browning, R. Heath-Brown, P. Salberger, Counting rational points on algebraic varieties. Duke Math. J. 132, 545–578 (2006)
H. Chen, Explicit uniform estimation of rational points I. Estimation of heights. J. Reine Angew. Math. 668, 59–88 (2012)
H. Chen, Explicit uniform estimation of rational points II. Hypersurface coverings. J. Reine Angew. Math. 668, 89–108 (2012)
S.K. Donaldson, Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)
J.-H. Evertse, R. Ferretti, Diophantine inequalities on projective varieties. Int. Math. Res. Notices 25, 1295–1330 (2002)
W. Fulton, Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, vol. 2 (Springer, Berlin, 1984)
D. Gieseker, Global moduli for surfaces of general type. Invent. Math. 43, 233–282 (1977)
J. Harris, I. Morrison, Moduli of Curves. Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998)
R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New-York, 1977)
R. Heath-Brown, The density of rational points on curves and surfaces. Ann. Math. 155, 553–595 (2002)
R. Heath-Brown, Imaginary quadratic fields with class group exponent 5. Forum Math. 20, 275–283 (2008)
W.V.D. Hodge, D. Pedoe, Methods of Algebraic Geometry, vol. 2 (Cambridge University Press, Cambridge, 1953)
S.J. Kleiman, Les théorèmes de finitude pour le foncteur de Picard, in Théorie des Intersections et Théorème de Riemann-Roch, Séminaire Géométrie Algébrique du Bois Marie 1966/67, SGA 6. Lecture Notes in Mathematic, vol. 225 (Springer, Berlin, 1971), pp. 616–666
O. Marmon, A generalization of the Bombieri-Pila determinant method. J. Math. Sci. 171, 736–744 (2010)
D. Mumford, Stability of projective varieties. Enseign. Math. 23, 39–110 (1977)
D. Mumford, J. Fogarty, Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 2, vol. 34 (Springer, Berlin, 1966)
S. Paul, G. Tian, CM stability and the generalized Futaki invariant I (2006). http://arxiv.org/abs/math/0605278
M.E. Rossi, N.V. Trung, G. Valla, Castelnuovo-mumford regularity and finiteness of Hilbert functions, in Commutative Algebra. Lecture Notes in Pure and Applied Mathematics, vol. 244 (Chapman Hall/CRC, Boca Raton, 2006), pp. 193–209
D. Rydh, Chow varieties, Master’s Thesis, Royal Institute of Technology, Stockholm, 2003
P. Salberger, On the density of rational and integral points on algebraic varieties. J. Reine Angew. Math. 606, 123–147 (2007)
P. Salberger, Uniform bounds for rational points on cubic hypersurfaces, in Arithmetic and Geometry. London Mathematical Society Lecture Note Series, vol. 420 (Cambridge University Press, Cambridge, 2015), pp. 401–421
P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique (Springer, Berlin, 1955)
E. Sernesi, Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften, vol. 334 (Springer, Berlin, 2007)
B. Sturmfels, Sparse Elimination Theory, in Computational Algebraic Geometry amd Commutative Algebra, ed. by D. Eisenbud, L. Robbiano (Cambridge University Press, Cambridge, 1993), p. 264
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Salberger, P. (2021). Chapter VI: On the Determinant Method and Geometric Invariant Theory. In: Peyre, E., Rémond, G. (eds) Arakelov Geometry and Diophantine Applications. Lecture Notes in Mathematics, vol 2276. Springer, Cham. https://doi.org/10.1007/978-3-030-57559-5_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-57559-5_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-57558-8
Online ISBN: 978-3-030-57559-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)