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Chapter VI: On the Determinant Method and Geometric Invariant Theory

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Arakelov Geometry and Diophantine Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2276))

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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.

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References

  1. E. Bombieri, J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59, 337–357 (1989)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Google Scholar 

  3. T.D. Browning, R. Heath-Brown, P. Salberger, Counting rational points on algebraic varieties. Duke Math. J. 132, 545–578 (2006)

    Article  MathSciNet  Google Scholar 

  4. H. Chen, Explicit uniform estimation of rational points I. Estimation of heights. J. Reine Angew. Math. 668, 59–88 (2012)

    MathSciNet  MATH  Google Scholar 

  5. H. Chen, Explicit uniform estimation of rational points II. Hypersurface coverings. J. Reine Angew. Math. 668, 89–108 (2012)

    MathSciNet  MATH  Google Scholar 

  6. S.K. Donaldson, Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)

    Article  MathSciNet  Google Scholar 

  7. J.-H. Evertse, R. Ferretti, Diophantine inequalities on projective varieties. Int. Math. Res. Notices 25, 1295–1330 (2002)

    Article  MathSciNet  Google Scholar 

  8. W. Fulton, Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 3, vol. 2 (Springer, Berlin, 1984)

    Google Scholar 

  9. D. Gieseker, Global moduli for surfaces of general type. Invent. Math. 43, 233–282 (1977)

    Article  MathSciNet  Google Scholar 

  10. J. Harris, I. Morrison, Moduli of Curves. Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998)

    Google Scholar 

  11. R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New-York, 1977)

    Google Scholar 

  12. R. Heath-Brown, The density of rational points on curves and surfaces. Ann. Math. 155, 553–595 (2002)

    Article  MathSciNet  Google Scholar 

  13. R. Heath-Brown, Imaginary quadratic fields with class group exponent 5. Forum Math. 20, 275–283 (2008)

    Article  MathSciNet  Google Scholar 

  14. W.V.D. Hodge, D. Pedoe, Methods of Algebraic Geometry, vol. 2 (Cambridge University Press, Cambridge, 1953)

    MATH  Google Scholar 

  15. 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

    Google Scholar 

  16. O. Marmon, A generalization of the Bombieri-Pila determinant method. J. Math. Sci. 171, 736–744 (2010)

    Article  MathSciNet  Google Scholar 

  17. D. Mumford, Stability of projective varieties. Enseign. Math. 23, 39–110 (1977)

    MathSciNet  MATH  Google Scholar 

  18. D. Mumford, J. Fogarty, Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 2, vol. 34 (Springer, Berlin, 1966)

    Google Scholar 

  19. S. Paul, G. Tian, CM stability and the generalized Futaki invariant I (2006). http://arxiv.org/abs/math/0605278

  20. 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

    Google Scholar 

  21. D. Rydh, Chow varieties, Master’s Thesis, Royal Institute of Technology, Stockholm, 2003

    Google Scholar 

  22. P. Salberger, On the density of rational and integral points on algebraic varieties. J. Reine Angew. Math. 606, 123–147 (2007)

    MathSciNet  MATH  Google Scholar 

  23. 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

    Google Scholar 

  24. P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique (Springer, Berlin, 1955)

    MATH  Google Scholar 

  25. E. Sernesi, Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften, vol. 334 (Springer, Berlin, 2007)

    Google Scholar 

  26. 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

    Google Scholar 

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Correspondence to Per Salberger .

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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

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