Abstract
Let K be a number field and consider a collection of equations \({{\rm{f}}_{\rm{1}}}({{\rm{x}}_{\rm{1}}}, \ldots ,{\rm{ }}{{\rm{x}}_n}) = {\rm{ }}{{\rm{f}}_{\rm{2}}}({{\rm{x}}_{\rm{1}}}, \ldots ,{\rm{ }}{{\rm{x}}_{\rm{n}}}) = \ldots = {\rm{ }}{{\rm{f}}_{\rm{r}}}\left( {{{\rm{x}}_{\rm{1}}}, \ldots ,{\rm{ }}{{\rm{x}}_{\rm{n}}}} \right){\rm{ }} = {\rm{ }}0\) where the f j are polynomials with coefficients in K. What are the solutions to these equations over K? Sometimes, every solution (a 1,…,a n ) ∈ K n necessarily satisfies further equations which are not algebraic consequences of our original collection. For instance, every solution over ℚ to x 31 + x 32 = 1 satisfies the additional equation x1x2 = 0. Of course, solutions over extensions L/K may fail to satisfy these further equations, e.g., the solution (-1, \(\sqrt[3]{2}\)) to the Fermat equation.
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References
S. A. Amitsur. Generic splitting fields of central simple algebras, Ann. of Math. (2), 62 (1955), 8–43.
W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 4, Springer-Verlag (Berlin, 1984).
F. Bogomolov and Y. Tschinkel, Density of rational points on Enriques surfaces, Math. Res. Letters, 5 (1998), 623–628.
F. Bogomolov and Y. Tschinkel, On the density of rational points on elliptic fibrations. J. reine angew. Math., 511 (1999), 87–93.
F. Bogomolov and Y. Tschinkel, Density of rational points on elliptic K3 surfaces, Asian J. of Math., 4 (2000), 351–368.
F. Campana. Remarques sur le revêtement universel des varietés kählériennes compactes, Bull. Soc. Math. France, 122 (1994), 255–284.
F. Campana, Special varieties and classification theory, math.AG/0110051.
J. L. Colliot-Thélène, A. N. Skorobogatov and P. Swinnerton-Dyer, Double fibres and double covers: paucity of rational points. Acta Arith., 79 (1997), 113–135.
A. Corti, A. Pukhlikov, M. Reid, Fano 3-fold hypersurfaces, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press (Cambridge, 2000), pp. 175–258.
O. Debarre, Fano varieties, this volume.
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983). 349–366;
G. Faltings, Erratum: Invent. Math., 75 (1984), 381.
G. Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2), 133 (1991), 549–576.
G. Frey and M. Jarden, Approximation theory and the rank of Abelian varieties over large algebraic fields, Proc. London Math. Soc., 28 (1974), 112–128.
G. Grant and E. Manduchi, Root numbers and algebraic points on elliptic surfaces with base P 1 Duke Math. J., 89 (1997), 413–422.
G. Grant and E. Manduchi. Root numbers and algebraic points on elliptic surfaces with elliptic base, Duke Math. J., 93 (1998), 479–486.
A. Grothendieck, Elements de géométrie algébrique IV: Etude locale des schémas et des morphismes de Schemas IV, Publ. Math. INES, vol. 32 (1967).
J. Harris and Y. Tschinkel, Rational points on quartics, Duke Math. J., 104 (2000), 477–500.
B. Hassett and Y. Tschinkel, Abelian fibrations and rational points on symmetric products. Intern. J. of Math., 11 (2000), 1163–1176.
B. Hassett and Y. Tschinkel, Density of integral points on algebraic varieties, in: Rational Points on Algebraic Varieties (E. Peyre, Yu. Tschinkel, eds.) Progress in mathematics, vol. 199, Birkhhäuser (2001), pp. 165–197.
V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry V, Encyclopaedia Math. Sci., vol. 47, Springer (1999), pp. 1–247.
K. Kodaira, On compact analytic surfaces I, II, III, Ann. of Math. (2), 71 (1960), 111–152;
K. Kodaira, On compact analytic surfaces I, II, III, Ann. of Math. (2), 77 (1963), 563–626;
K. Kodaira, On compact analytic surfaces I, II, III, Ann. of Math. (2), 78 (1963), 1–40.
J. Kollár, Higher direct images of dualizing sheaves II, Ann. of Math. (2), 124 (1986), no. 1, 171–202.
J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32, Springer (1996).
S. J. Kovács, The cone of curves of a K3 surface, Math. Ann., 300 (1994), 681–691.
S. Lang, Abelian Varieties, Interscience Tracts in Pure and Applied Mathematics, Number 7 (1959).
S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math., 80 (1958), 659–684.
E. Looijenga and C. Peters, Torelli theorems for Kahler K3 surfaces, Compositio Math., 42 (1980/81), 145–186.
T. Miyake, Modular forms, Springer (1989).
S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus 11, Algebraic geometry (Tokyo/Kyoto, 1982), LNM 1016, Springer (1983), pp. 334–353.
S. Mukai, On the moduli space of bundles on K3 surfaces I, Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math., 11, Tata Inst. Fund. Res. (Bombay, 1987), pp. 341–413.
A. P. Ogg, Cohomology of abelian varieties over function fields, Ann. of Math. (2), 76 (1962), 185–212.
Z. Ran. Hodge theory and deformations of maps, Compositio Math., 97 (1995), 309–328.
P. Roquette, On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras. Math. Ann., 150 (1963), 411–439.
B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math., 96 (1974), 602–639.
J. P. Serre, A course in arithmetic, Graduate Texts In Mathematics, vol. 7, Springer (1973).
J. P. Serre, Lectures on the Mordell-Weil Theorem, 2nd edition, Vieweg (Braunschweig, 1990).
I. R. Shafarevich, Principal homogeneous spaces defined over a function field (Russian) Trudy Mat. Inst. Stehlov., 64 (1961), 316–346;
I. R. Shafarevich, English translation: Amer. Math. Soc. Transl. (2), 37 (1964), 85–114.
I. R. Shafarevich (ed.) Algebraic surfaces Proceedings of the Steklov Institute of Mathematics, No. 75 (1965) American Mathematical Society (Providence, R.I., 1965).
T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan, 24 (1972), 20–59.
T. Shioda and H. Inose, On singular K3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten (Tokyo, 1977), pp. 119–136.
J. Silverman, Rational points on K3 surfaces: a new canonical height, Invent. Math,, 105 (1991). 347–373.
H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z., 189 (1985), 507–513.
A. Weil. Arithmétique et géometrie sur les variétés algébriques, in: Actualités Sci. Indust., 206, Hermann (Paris, 1936), pp. 3–16; reprinted in: Oeuvres Scientifiques Vol. I, Springer (1980), pp. 87–100.
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Hassett, B. (2003). Potential Density of Rational Points on Algebraic Varieties. In: Böröczky, K., Kollár, J., Szamuely, T. (eds) Higher Dimensional Varieties and Rational Points. Bolyai Society Mathematical Studies, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05123-8_8
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