Duke Mathematical Journal

Hypersurfaces in projective schemes and a moving lemma

Ofer Gabber, Qing Liu, and Dino Lorenzini

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Abstract

Let X / S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H / S of X / S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X / S containing a given closed subscheme C and intersecting properly a closed set F .

Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z S , Pic ( Z ) is a torsion group. This condition is satisfied if R is the ring of integers of a number field or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1 -cycles on a regular scheme X quasi-projective and flat over S . We also show the existence of a finite surjective S -morphism to P S d for any scheme X projective over S when X / S has all its fibers of a fixed dimension d .

Article information

Source
Duke Math. J., Volume 164, Number 7 (2015), 1187-1270.

Dates
Received: 16 July 2011
Revised: 9 July 2014
First available in Project Euclid: 14 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1431608068

Digital Object Identifier
doi:10.1215/00127094-2877293

Mathematical Reviews number (MathSciNet)
MR3347315

Zentralblatt MATH identifier
06455741

Subjects
Primary: 14A15: Schemes and morphisms 14C25: Algebraic cycles 14D06: Fibrations, degenerations 14D10: Arithmetic ground fields (finite, local, global) 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]

Keywords
avoidance lemma Bertini-type theorem hypersurface moving lemma Noether normalization $1$-cycle pictorsion quasisection rational equivalence zero locus of a section

Citation

Gabber, Ofer; Liu, Qing; Lorenzini, Dino. Hypersurfaces in projective schemes and a moving lemma. Duke Math. J. 164 (2015), no. 7, 1187--1270. doi:10.1215/00127094-2877293. https://projecteuclid.org/euclid.dmj/1431608068


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References

  • [1] S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348.
  • [2] A. Altman and S. Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Math. 146, Springer, New York, 1970.
  • [3] S. Anantharaman, Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension $1$, Bull. Soc. Math. France Mém. 33 (1973), 5–79.
  • [4] M. Artin, “Algebraic spaces” in A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Math. Monogr. 3, Yale Univ. Press, New Haven, Conn, 1971.
  • [5] A. Bialynicki-Birula, J. Browkin, and A. Schinzel, On the representation of fields as finite unions of subfields, Colloq. Math. 7 (1959), 31–32.
  • [6] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. 3, Springer, Berlin, 1990.
  • [7] N. Bourbaki, Commutative Algebra: Chapters 1–7, translated from the French, reprint of the 1989 English translation. Elem. of Math. Springer, Berlin, 1998.
  • [8] K. Buzzard, Smooth proper scheme over Z, available at http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z (accessed 5 January 2015).
  • [9] C. Chevalley, Sur la théorie des variétés algébriques, Nagoya Math. J. 8 (1955), 1–43.
  • [10] T. Chinburg, G. Pappas, and M. J. Taylor, Finite morphisms from curves over Dedekind rings to ${\mathbb{P}}^{1}$, preprint, arXiv:0902.2039v2 [math.AG].
  • [11] T. Chinburg, L. Moret-Bailly, G. Pappas, and M. J. Taylor, Finite morphisms to projective space and capacity theory, preprint, arXiv:1201.0678v2 [math.AG].
  • [12] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.
  • [13] L. van den Dries and A. Macintyre, The logic of Rumely’s local-global principle, J. Reine Angew. Math. 407 (1990), 33–56.
  • [14] A. de Jong and F. Oort, On extending families of curves, J. Algebraic Geom. 6 (1997), 545–562.
  • [15] A. de Jong and M. Pikaart, “Moduli of curves with non-abelian level structure” in The Moduli Space of Curves (Texel Island, 1994), Progr. Math. 129, Birkhäuser Boston, 1995, 483–509.
  • [16] S. Diaz and D. Harbater, Strong Bertini theorems, Trans. Amer. Math. Soc. 324 (1991), 73–86.
  • [17] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995.
  • [18] J. Emsalem, Projectivité des schémas en courbes sur un anneau de valuation discrète, Bull. Soc. Math. France 101 (1974), 255–263.
  • [19] D. Estes and R. Guralnick, Module equivalences: Local to global when primitive polynomials represent units, J. Algebra 77 (1982), 138–157.
  • [20] D. Ferrand, Trivialisation des modules projectifs: La méthode de Kronecker, J. Pure Appl. Algebra 24 (1982), 261–264.
  • [21] D. Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), 553–585.
  • [22] G. Frey and M. Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112–128.
  • [23] W. Fulton, Intersection Theory, Ergeb. Math. 2, Springer, Berlin, 1984.
  • [24] W. Fulton and S. Lang, Riemann-Roch Algebra, Grund. Math. Wiss. 277, Springer, New York, 1985.
  • [25] O. Gabber, Q. Liu, and D. Lorenzini, The index of an algebraic variety, Invent. Math. 192 (2013), 567–626.
  • [26] I. Goldbring and M. Masdeu, Bézout domains and elliptic curves, Comm. Algebra 36 (2008), 4492–4499.
  • [27] O. Goldman, On a special class of Dedekind domains, Topology 3, suppl. 1 (1964), 113–118.
  • [28] B. Green, Geometric families of constant reductions and the Skolem property, Trans. Amer. Math. Soc. 350 (1998), 1379–1393.
  • [29] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, Institut Hautes Études Sci. Publ. Math. 4 (Chapter 0, 1–7, and I, 1–10), 8 (II, 1–8), 11 (Chapter 0, 8–13, and III, 1–5), 17 (III, 6–7), 20 (Chapter 0, 14–23, and IV, 1), 24 (IV, 2–7), 28 (IV, 8–15), and 32 (IV, 16–21), 1960–1967.
  • [30] A. Grothendieck and J. Dieudonné, Éléments de Géométrie Algébrique, I, Grund. Math. Wiss. 166, Springer, Berlin, 1971.
  • [31] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • [32] R. Guralnick, D. Jaffe, W. Raskind, and R. Wiegand, On the Picard group: Torsion and the kernel induced by a faithfully flat map, J. Algebra 183 (1996), 420–455.
  • [33] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [34] J.-P. Jouanolou, “Une suite exacte de Mayer-Vietoris en K-théorie algébrique” in Algebraic K-theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973, 293–316.
  • [35] J.-P. Jouanolou, “Quelques calculs en K-Théorie des schémas” in Algebraic K-Theory, I: Higher K-Theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973, 317–335.
  • [36] J.-P. Jouanolou, Le formalisme du résultant, Adv. Math. 90 (1991), 117–263.
  • [37] I. Kaplansky, Commutative Rings, Rev. ed., Univ. of Chicago Press, Chicago, 1974.
  • [38] K. Kedlaya, More étale covers of affine spaces in positive characteristic, J. Algebraic Geom. 14 (2005), 178–192.
  • [39] S. Kleiman, Misconceptions about $K_{X}$, Enseign. Math. 25 (1979), 203–206.
  • [40] S. Kleiman and A. Altman, Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 (1979), 775–790.
  • [41] J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), 235–268.
  • [42] W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche, VIII: Multiplikativ abgeschlossene Systeme von endlichen Idealen, Math. Z. 48 (1943), 533–552.
  • [43] G. Laumon and L. Moret-Bailly, Champs Algébriques, Ergeb. der Math. Grenzgeb., 3. Folge, 39, Springer, Berlin, 2000.
  • [44] D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128.
  • [45] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Grad. Texts in Math. 6, Oxford Univ. Press, New York, 2006.
  • [46] J. Milne, “Abelian varieties” in Arithmetic Geometry (Storrs, Conn., 1984), Springer, New York, 1986, 103–150.
  • [47] S. McAdam, “Finite coverings by ideals” in Ring Theory (Oklahoma, 1973), Lecture Notes in Pure and Appl. Math. 7, Dekker, New York, 1974, 163–171.
  • [48] B. McDonald and W. Waterhouse, Projective modules over rings with many units, Proc. Amer. Math. Soc. 83 (1981), 455–458.
  • [49] L. Moret-Bailly, “Points entiers des variétés arithmétiques” in Séminaire de Théorie des Nombres (Paris 1985–86), Progr. Math. 71, Birkhäuser, Boston, 1987, 147–154.
  • [50] L. Moret-Bailly, Groupes de Picard et problèmes de Skolem, I, Ann. Sci. École Norm. Sup. (4) 22 (1989), 161–179.
  • [51] D. Mumford, Lectures on Curves on an Algebraic Surface, Ann. of Math. Stud. 59, Princeton Univ. Press, Princeton, 1966.
  • [52] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgebiete (2) 34, Springer, Berlin, 1994.
  • [53] M. P. Murthy and R. Swan, Vector bundles over affine surfaces, Invent. Math. 36 (1976), 125–165.
  • [54] M. Nagata, Local Rings, Interscience Tracts in Pure and Appl. Math. 13, Wiley, New York, 1962.
  • [55] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.
  • [56] M. Nishi, On the imbedding of a nonsingular variety in an irreducible complete intersection, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 29 (1955), 172–187.
  • [57] G. Pearlstein and Ch. Schnell, “The zero locus of the infinitesimal invariant” in Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Fields Inst. Commun. 67, Springer, New York, 2013, 589–602.
  • [58] R. Piene, Courbes sur un trait et morphismes de contraction, Math. Scand. 35 (1974), 5–15.
  • [59] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), 1099–1127.
  • [60] B. Poonen, Smooth hypersurface sections containing a given subscheme over a finite field, Math. Res. Lett. 15 (2008), 265–271.
  • [61] J. Roberts, “Chow’s moving lemma,” Appendix 2 to “Motives” in Algebraic Geometry (Oslo, 1970), Wolters-Noordhoff, Groningen, 1972, 53–82.
  • [62] M. Rosen, $S$-units and $S$-class group in algebraic function fields, J. Algebra 26 (1973), 98–108.
  • [63] R. Rumely, Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math. 368 (1986), 127–133.
  • [64] P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282–301.
  • [65] S. Schröer, On non-projective normal surfaces, Manuscripta Math. 100 (1999), 317–321.
  • [66] E. Sernesi, Deformations of algebraic schemes, Grund. Math. Wiss 334, Springer, Berlin, 2006.
  • [67] J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, fasc. 2, exp. 23, Secrétariat mathématique, Paris.
  • [68] J.-P. Serre, “Lettre à M. Tsfasman,” in Journées Arithmétiques (Luminy, 1989), Astérisque 198–200 (1991), 351–353.
  • [69] H. Sumihiro, A theorem on splitting of algebraic bundles and its applications, Hiroshima Math. J. 12 (1982), 435–452.
  • [70] R. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277.
  • [71] I. Swanson and C. Huneke, Integral Closure of Ideals, Rings, and Modules, Lond. Math. Soc. Lecture Note Ser. 336, Cambridge Univ. Press, Cambridge, 2006.
  • [72] R. Swan and J. Towber, A class of projective modules which are nearly free, J. Algebra 36 (1975), 427–434.
  • [73] A. Tamagawa, Unramified Skolem problems and unramified arithmetic Bertini theorems in positive characteristic, Doc. Math. Extra Vol. (2003), 789–831.
  • [74] A. Thorup, “Rational equivalence theory on arbitrary Noetherian schemes” in Enumerative Geometry (Sitges, 1987), Lecture Notes in Math. 1436, Springer, Berlin, 1990, 256–297.
  • [75] W. van der Kallen, The $K_{2}$ of rings with many units, Ann. Sci. École Norm. Sup. (4) 10 (1977), 473–515.
  • [76] R. M. W. Wood, Polynomial maps of affine quadrics, Bull. Lond. Math. Soc. 25 (1993), 491–497.