Abstract
We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.
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References
Atiyah, M.F., MacDonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1969)
Bădescu, L.: Algebraic surfaces. Universitext. Springer, New York (2001). Translated from the 1981 Romanian original by Vladimir Masek and revised by the author
Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. \(p\). II. In: Complex analysis and algebraic geometry, pp. 23–42. Iwanami Shoten, Tokyo (1977)
Bosch, S.: Meromorphic functions on proper rigid analytic varieties. In: Seminar on number theory, 1982–1983 (Talence, 1982/1983), Exp. No. 34. Univ. Bordeaux I, Talence (1983)
Bosch, S.: Minimal models of rigid analytic spaces (2008, preprint). http://wwwmath.uni-muenster.de/sfb/about/publ/heft378.pdf
Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences. A systematic approach to rigid analytic geometry), vol. 261. Springer, Berlin (1984)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Formal and rigid geometry. III. The relative maximum principle. Math. Ann. 302(1), 1–29 (1995)
Conrad, B.: Modular curves and rigid-analytic spaces. Pure Appl. Math. Q. 2(1, part 1), 29–110 (2006)
Conrad, B.: Relative ampleness in rigid geometry. Ann. Inst. Fourier (Grenoble) 56(4), 1049–1126 (2006)
Conrad, B.: Moishezon spaces in rigid geometry (2010, preprint)
Cossec, F.R., Dolgachev, I.V.: Enriques surfaces. I. Progress in Mathematics, vol. 76. Birkhäuser Boston Inc., Boston (1989)
Demazure, M., Grothendieck, A.: Schéme en groupes, Lecture Notes in Mathematics, vol. 151, 152, 153. Springer-Verlag, Berlin (1970). Séminaire de Géométrie Algébrique du Bois Marie 1962–1964 (SGA 3) Avec la collaboration de M. Artin, J. E. Bertin, P. Gabriel, M. Raynaud, et J. P. Serre.
Dolgačev, I.V.: The Euler characteristic of a family of algebraic varieties. Mat. Sb. (N.S.) 89(131), 297–312, 351 (1972)
Grothendieck, A.: Revêtements étales et groupe fondamental. Springer-Verlag, Berlin, : Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1). Augmenté de deux exposés de M, Raynaud, Lecture Notes in Mathematics, vol. 224 (1971)
Gruson, L.: Fibré vectoriels sur un polydisque ultramétrique. Ann. Sci. École Norm. Sup 1(4), 45–89 (1968)
Ito, R., Katsura, T.: A note on wild fibers of elliptic surfaces. J. Pure Appl. Algebra 109(2), 127–141 (1996)
de Jong, J., van der Put, M.: Étale cohomology of rigid analytic spaces. Doc. Math. 1(1), 1–56 (1996) (electronic)
Katsura, T.: Multicanonical systems of elliptic surfaces in small characteristics. Compos. Math. 97(1–2), 119–134 (1995) (special issue in honour of Frans Oort)
Katsura, T., Ueno, K.: On elliptic surfaces in characteristic \(p\). Math. Ann. 272(3), 291–330 (1985)
Katsura, T., Ueno, K.: Multiple singular fibres of type \({\bf G}_{a}\) of elliptic surfaces in characteristic \(p\). In: Algebraic and topological theories (Kinosaki, 1984), pp. 405–429. Kinokuniya, Tokyo (1986)
Kawazoe, M.: A criterion for algebraicity of certain analytic elliptic surfaces. Math. Ann. 303(4), 771–779 (1995)
Kisin, M.: Local constancy in families of non-abelian Galois representations. Math. Z. 233(2), 347–363 (2000)
Kodaira, K.: On the structure of compact complex analytic surfaces. I. Am. J. Math. 86, 751–798 (1964)
Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schr. Math. Inst. Univ. Münster (2) 7, iv+72 (1974)
Lichtenbaum, S.: Curves over discrete valuation rings. Am. J. Math. 90, 380–405 (1968)
Liu, Q.: Algebraic geometry and arithmetic curves. Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, Oxford (2002). Translated from the French by Reinie Erné, Oxford Science Publications
Liu, Q., Lorenzini, D., Raynaud, M.: Néron models, Lie algebras, and reduction of curves of genus one. Invent. Math. 157(3), 455–518 (2004)
Mehlmann, F.: Ein Beweis für einen Satz von Raynaud über flache Homomorphismen affinoider Algebren. Schr. Math. Inst. Univ. Münster (2) 19, iv+112 (1981)
Mitsui, K.: Bimeromorphic geometry of rigid analytic surfaces. Int. J. Math. 22(4), 483–513 (2011)
Mumford, D.: Pathologies of modular algebraic surfaces. Am. J. Math. 83, 339–342 (1961)
Mumford, D., Forgarty, J., Kirwan, F.: Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, 3rd edn. Springer, Berlin (1994)
Oguiso, K.: An elementary proof of the topological Euler characteristic formula for an elliptic surface. Comment. Math. Univ. St. Paul. 39(1), 81–86 (1990)
Oort, F.: Sur le schéma de Picard. Bull. Soc. Math. France 90, 1–14 (1962)
Raynaud, M.: Spécialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38, 27–76 (1970)
Rudakov, A.N., Šafarevič, I.R.: Inseparable morphisms of algebraic surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 40(6), 1269–1307, 1439 (1976)
Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106, 2nd edn. Springer, Dordrecht (2009)
Tate, J.: A review of non-Archimedean elliptic functions. In: Elliptic curves, modular forms,& Fermat’s last theorem (Hong Kong, 1993). Ser. Number Theory, I, pp. 162–184. Int. Press, Cambridge (1995)
Ueno, K.: Compact rigid analytic spaces with special regard to surfaces. In: Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, pp. 765–794. North-Holland, Amsterdam (1987)
Acknowledgments
The author thanks the referee for a lot of helpful comments. He thanks Professor Fumiharu Kato for his support and Professor Tadao Oda for his comments on the manuscript. This work was supported by the Grant-in-Aid for Japan Society for the Promotion of Science Fellows (21-1111).
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Appendix
Appendix
1.1 Local properties of elliptic fibrations with a section
In this subsection, we show that any two smooth elliptic fibrations with the same \(j\)-invariant are isomorphic over a sufficiently small admissible open neighborhood of a given point. In particular, we can apply our logarithmic transformations to a fiber of a smooth elliptic fibration whenever the absolute value of the \(j\)-invariant of the fiber is greater than one.
Proposition 8
Let \((X,C,\pi )\) be a smooth elliptic fibration. Take a point \(q\) on \(C.\) Then there exists an admissible open subset \(U\) of \(C\) containing \(q\) such that the restriction \(\pi |_{\pi ^{-1}(U)}:\pi ^{-1}(U)\rightarrow U\) admits a section.
Proof
Take a point \(q^{\prime }\) on the fiber \(\pi ^{-1}(q).\) Take an admissible affinoid open subset \(U\) of \(C\) and an admissible affinoid open subset \(V\) of \(X\) satisfying the conditions that \(q^{\prime }\in V\) and \(V\subset \pi ^{-1}(U).\) After shrinking \(U\) and \(V,\) we may assume that there exists a \(U\)-isomorphism \(V\cong (\pi ^{-1}(q)\cap V)\times U.\) This fact follows from Proposition 1.2 in [22] for the case where \(\fancyscript{Z}\) in the proposition is empty. Thus, the restriction \(\pi |_V:V\rightarrow U\) admits a section \(\sigma .\) The composite \(\iota \circ \sigma \) of the section \(\sigma \) and the canonical open \(U\)-immersion \(\iota :V\rightarrow \pi ^{-1}(U)\) is a section of the restriction \(\pi |_{\pi ^{-1}(U)}:\pi ^{-1}(U)\rightarrow U.\) Thus, the admissible open subset \(U\) is a desired one. \(\square \)
Proposition 9
Let \(\nu :B\rightarrow C\) be a finite morphism between smooth affinoid spaces that are curves. Let \((X,C,\pi )\) be an elliptic fibration. Assume that the reductions of all the fibers of \(\pi \) are isomorphic to elliptic curves. Suppose that \(\nu \) factors through \(\pi .\) Then the elliptic fibration is the analytification of an algebraic one.
Proof
We take a morphism \(\sigma :B\rightarrow X\) such that the equality \(\pi \circ \sigma =\nu \) holds. Since \(\sigma \) is proper (Proposition 5 in [6, 9.6.2] and Proposition 4 in [6, 9.6.2]), the proper mapping theorem (Proposition 3 in [6, 9.6.3]) shows that the image \(E\) of \(\sigma \) is a closed analytic subset of \(X.\) We endow \(E\) with the reduced structure. Then \(E\) is a divisor on \(X.\) By the same method as in the proof of Theorem 6, we can show that the line bundle \(\fancyscript{O}_X(E)\) is relatively ample for \(\pi .\) Thus, the fibration \(\pi \) is projective (Theorem 3.2.7 in [9]). Therefore, the proposition follows from the relative GAGA theorems [24]. \(\square \)
We define the \(j\)-invariant for an elliptic fibration with a section to be the \(j\)-invariant for the algebraization of the fibration.
Proposition 10
Let \(C\) be a smooth affinoid space that is a curve. Let \((X,C,\pi )\) and \((X^{\prime },C,\pi ^{\prime })\) be smooth elliptic fibrations with sections. Suppose that the two elliptic fibrations have the same \(j\)-invariant. Then, for any point \(q\) on \(C,\) there exists an admissible affinoid open neighborhood \(U\) of \(q\) such that the restrictions \(\pi ^{-1}(U)\) and \((\pi ^{\prime })^{-1}(U)\) are isomorphic over \(U\) in the category of rigid analytic spaces.
Proof
Proposition 9 shows that the fibrations \((X,C,\pi )\) and \((X^{\prime },C,\pi ^{\prime })\) are the analytifications of algebraic ones. In the following, we use the same letters \(X, X^{\prime },\) and \(C\) to denote their algebraizations. Put \(A:=\fancyscript{O}_C(C).\) To show the proposition, we have only to show that \(X\times _A\fancyscript{O}_{C,q}\) is isomorphic to \(X^{\prime }\times _A\fancyscript{O}_{C,q}\) over \(\fancyscript{O}_{C,q}.\) Since \(\fancyscript{O}_{C,q}\) is a strictly Henselian discrete valuation ring (Lemma 2.1.1 (1) in [17]), the proposition follows from the following proposition. \(\square \)
Proposition 11
Let \(R\) be a strictly Henselian discrete valuation ring. Let \(X\) and \(X^{\prime }\) be elliptic curves over \(R.\) Suppose that the two elliptic curves have the same \(j\)-invariant. Then \(X\) and \(X^{\prime }\) are isomorphic over \(R.\)
Proof
Let \(L\) be the field of fractions of \(R.\) We take a minimal Weierstrass form
of \(X\) (resp. \(X^{\prime }\)) over \(R\) (see [36, VII.1]). We define \(b_2,b_4,b_6,b_8,c_4\in R\) in the following way:
The discriminant \(\Delta \) and the \(j\)-invariant of \(X\) are given by
and
respectively. We define \(b_2^{\prime },b_4^{\prime },b_6^{\prime },b_8^{\prime },c_4^{\prime }\in R\) for \(X^{\prime }\) in the same way. The discriminant \(\Delta ^{\prime }\) and the \(j\)-invariant \(j^{\prime }\) of \(X^{\prime }\) are also given in the same way.
Since \(j=j^{\prime },\) there exists a finite separable extension \(M/L\) such that \(X\times _RM\) and \(X^{\prime }\times _RM\) are isomorphic over \(M.\) The isomorphism is given by
where \(r_0,s_0,t_0,u_0\in M\) (Proposition III.3.1 (b) in [36]). We have the following equalities (Table 3.1 in [36, p. 45]):
Let \(S\) be the integral closure of \(R\) in \(M.\) Since \(\Delta ,\Delta ^{\prime }\in R^\times ,\) the equality (10) shows \(u_0\in S^\times .\) The equality (9) shows \(3r_0\in S.\) The equality (8) shows \(4r_0\in S.\) Then \(r_0\in S.\) The equality (4) shows \(s_0\in S.\) The equality (7) shows \(t_0\in S.\) Therefore, to show the proposition, we have only to show \(r_0,s_0,t_0,u_0\in L.\) Let \(p\) be the characteristic of the residue field of \(R.\)
Case (I): \(p\not =2,3.\) Since \(\Delta ,\Delta ^{\prime }\in R^\times ,\) the equality (10) gives \(u_0^{12}\in R^\times .\) Since \(R\) is strictly Henselian, we obtain \(u_0\in R.\) The equality (3) shows \(s_0\in R.\) The equality (4) shows \(r_0\in R.\) The equality (5) shows \(t_0\in R.\) Therefore, the proposition holds. We remark that this argument works whenever the characteristic of \(L\) is different from \(2\) and \(3.\)
Referring to the equalities (3–7), we define \(f_i\in R[r,s,t,u]\) in the following way:
Case (II): \(p=3.\) Replacing \(y\) by \(y-(a_1x+a_3)/2\) and replacing \(y^{\prime }\) in the same way, we may assume that the equalities \(a_1=a_3=a_1^{\prime }=a_3^{\prime }=0\) hold. The equality (3) shows \(s_0=0.\) Thus, the equality (4) shows the following: \(a_2\in R^\times \) if and only if \(a_2^{\prime }\in R^\times .\) First, we consider the case where \(a_2\in R^\times .\) Put \(T:=R[r,s,t,u].\) Let \(\psi :R\rightarrow T\) be the \(R\)-algebra structure homomorphism. Let \(\phi :T\rightarrow S\) be the \(R\)-algebra homomorphism given by
Then \((f_1,f_2,f_3,f_4,f_6)\subset \text{ Ker}\phi .\) Let \(\mathfrak m \) be the maximal ideal of \(R.\) Since \(S\) is integral over \(R,\) there exists a prime ideal \(\mathfrak n \) of \(S\) such that \((\phi \circ \psi )^{-1}(\mathfrak n )=\mathfrak m .\) Put \(\mathfrak p :=\phi ^{-1}(\mathfrak n ), T^{\prime }:=T_{\mathfrak p }/(f_1,f_2,f_3,f_4),\) and \(d:=\det (\partial (f_1,f_2,f_3,f_4)/\partial (r,s,t,u)).\) The equality
shows \(\phi (d)\in S^\times ,\) which implies \(d\in T_{\mathfrak p }^\times .\) Thus, the \(R\)-algebra \(T^{\prime }\) is finite and étale over \(R.\) Since \(R\) is strictly Henselian, the \(R\)-algebra structure homomorphism \(R\rightarrow T^{\prime }\) is an isomorphism. This concludes that \(r_0,s_0,t_0,u_0\in R.\) Therefore, the proposition holds. Next, we consider the case where \(a_2\not \in R^\times .\) The assumptions \(\Delta ,\Delta ^{\prime }\in R^\times \) show \(a_4,a_4^{\prime }\in R^\times .\) Since the equality
holds, this case can be shown by a similar argument for \((f_1,f_3,f_4,f_6).\)
Case (III): \(p=2.\) Replacing \(x\) by \(x+a_2\) and replacing \(x^{\prime }\) in the same way, we may assume that the equalities \(a_2=a_2^{\prime }=0\) hold. The equality (3) shows the following: \(a_1\in R^\times \) if and only if \(a_1^{\prime }\in R^\times .\) First, we consider the case where \(a_1\in R^\times .\) Since the equality
holds, this case can be shown by a similar argument for \((f_1,f_2,f_3,f_4).\) Next, we consider the case where \(a_1\not \in R^\times .\) The assumptions \(\Delta ,\Delta ^{\prime }\in R^\times \) show \(a_3,a_3^{\prime }\in R^\times .\) Since the equality
holds, this case can be shown by a similar argument for \((f_2,f_3,f_4,f_6).\) \(\square \)
1.2 Multiple fibers of elliptic fibrations
In this subsection, we provide steps necessary for the proof of Proposition 1. The analogous studies for elliptic surfaces are given by Proposition 4 in [3]. For completeness, we prove the following lemmas and propositions for algebraic elliptic fibrations.
Let \((X,C,\pi )\) be a relatively minimal algebraic elliptic fibration. By \(f_\eta :X_\eta \rightarrow C_\eta \) we denote the generic fiber of \(f.\) Take a closed point \(q\) on \(C.\) Put \(X_q:=\pi ^{-1}(q).\) We write \(X_q=\sum _{i\in I}d_i\Gamma _i\) where \(\{\Gamma _i\}_{i\in I}\) is the set of the irreducible components of the reduction of \(X_q.\) Let \(m\) be the multiplicity of \(X_q.\) We define an effective divisor \(D\) on \(X\) by \(D:=X_q/m.\) Let \(\nu \) be the order of the normal bundle \(\fancyscript{N}_{D/X}\) of \(D\) in \(\text{ Pic}D.\) Let \(E\) be a divisor on \(X\) satisfying \(0<E\le X_q.\) We regard \(E\) as a scheme over the residue field of \(\fancyscript{O}_{C,q}.\) Let \(\imath :E\rightarrow X_q\) be the canonical closed immersion. For any line bundle \(\fancyscript{L}\) on \(E,\) we abbreviate \(\imath _*\fancyscript{L}\) to \(\fancyscript{L}\) for simplicity. We define the degree of a line bundle \(\fancyscript{L}\) on \(E\) by the equality
(Definition 7.3.29 in [26]). The map \(\deg :\text{ Pic}E\rightarrow {\mathbb Z }\) is a group homomorphism (Lemma 7.3.30 (b) in [26]). By \(E_1\cdot E_2\) we denote the intersection number of two divisors \(E_1\) and \(E_2\) on \(X\) whose supports are contained in the special fiber of \(\pi .\) If the inequalities \(0<E_2\le X_q\) hold, then the equality
holds (Theorem 9.1.12 in [26]).
Lemma 13
Let \(\fancyscript{L}\) be a line bundle on \(D.\) Assume that the equality \(\deg \fancyscript{L}|_{\Gamma _i}=0\) holds for any \(i\in I.\) Then the equality \(\deg \fancyscript{L}|_E=0\) holds for any divisor \(E\) on \(X\) satisfying \(0<E\le D.\)
Proof
If \(E\) is a prime divisor, then there exists an element \(i\in I\) such that the equality \(E=\Gamma _i\) holds. Thus, we may assume that there exist two non-zero effective divisors \(E_1\) and \(E_2\) such that the equality \(E=E_1+E_2\) holds. Tensoring \(\fancyscript{L}\) with the short exact sequence
we obtain a short exact sequence
The two short exact sequences give the equalities
and
Since the equalities
and
hold, the above two equalities give the equality
Thus, we obtain the desired equality by induction. \(\square \)
Lemma 14
Let \(\fancyscript{L}\) be a line bundle on \(D.\) Assume that the equality \(\deg \fancyscript{L}|_{\Gamma _i}=0\) holds for any \(i\in I.\) Then the following conditions are equivalent.
-
1.
The line bundle \(\fancyscript{L}\) is trivial.
-
2.
The equality \(h^0(\fancyscript{L})=1\) holds.
-
3.
The line bundle \(\fancyscript{L}\) admits a non-zero global section.
Proof
The implication Condition 1 \(\Rightarrow \) Condition 3 and the implication Condition 2 \(\Rightarrow \) Condition 3 are clear. Let us show the implications Condition 3 \(\Rightarrow \) Conditions 1 and 2. Assume that Condition 3 is satisfied. Let \(h\) be a non-zero global section of \(\fancyscript{L}.\) Let \(D_1\) be the maximal effective divisor such that \(D_1\le D\) and \(h|_{D_1}=0.\) Put \(D_2:=D-D_1.\) Lemma 13 gives the equality \(\deg \fancyscript{L}|_{D_2}=0.\) Thus, the equalities
hold. Let \(\phi :\fancyscript{O}_{D_2}\rightarrow \fancyscript{O}_X(-D_1)|_{D_2}\otimes \fancyscript{L}\) be the canonical homomorphism induced by \(h.\) Put \(\fancyscript{F}:=\text{ Coker}\phi .\) Then the above equalities and the short exact sequence
give the equality
Since the support of \(\fancyscript{F}\) is at most zero-dimensional, the inequality \(\chi (\fancyscript{F})\ge 0\) holds. Since the equality \(D_2\not =0\) holds by assumption, Theorem 9.1.23 in [26] gives the equality \(D_1=0.\) Thus, we obtain the equality \(\chi (\fancyscript{F})=0,\) which implies \(\fancyscript{F}=0.\) Therefore, the global section \(h\) is nowhere-vanishing. The fact implies that Conditions 1 and 2 are satisfied. \(\square \)
Lemma 15
The equality \(\deg \fancyscript{N}_{D/X}^{\otimes n}|_{\Gamma _i}=0\) holds for any integer \(n\) and any \(i\in I.\)
Proof
The equality follows from the equalities
\(\square \)
For each integer \(n,\) put
Lemma 16
For any integer \(n\) satisfying \(0<n\le m,\) the equality
holds. For any integer \(n,\) the equalities
hold.
Proof
Let \(n\) be an integer.Since the equalities
hold, we obtain the equality
If \(n>0,\) then there exists a short exact sequence
Thus, if \(0<n<m,\) then we obtain the equality
Therefore, by induction on \(n,\) we obtain the equality \(\chi (\fancyscript{O}_{nD})=n\chi (\fancyscript{O}_D)\) for any integer \(n\) satisfying \(0<n\le m.\) The constancy of the Euler characteristics of \(C\)-flat coherent sheaves on a proper family of schemes over \(C\) gives the equality \(\chi (\fancyscript{O}_{X_q})=0.\) Since the equality \(X_q=mD\) holds by definition, the equality \(\chi (\fancyscript{O}_{mD})=0\) holds. Thus, the equality \(\chi (\fancyscript{O}_{nD})=0\) holds for any integer \(n\) satisfying \(0<n\le m.\) Therefore, the equality \(\chi (\fancyscript{N}_n)=0\) holds for any integer \(n.\) Since \(D\) is of dimension one, we obtain the first and second equalities. Since there exists an isomorphism
for any integer \(n,\) the last equality follows from Lemmas 15 and 14. \(\square \)
Lemma 17
The fiber \(X_q\) is tamely ramified if and only if the equality \(h^1(\fancyscript{O}_{X_q})=1\) holds.
Proof
Since \(R^2f_*\fancyscript{O}_X=0,\) the lemma follows from Theorem 5.3.20 (d) in [26]. \(\square \)
Proposition 12
For each positive integer \(n\) satisfying \(0<n\le m,\) put
Then the following statements hold.
-
1.
The equality \(F(1)=1\) holds, and the function \(F(n)\) is non-decreasing.
-
2.
For any positive integer \(n,\) the inequality \(F(n)>1\) holds if and only if the inequality \(n>\nu \) holds.
In particular, the fiber \(X_q\) is tamely ramified if and only if the equality \(\nu =m\) holds.
Proof
Lemma 16 for \(n=0\) gives the equality \(F(1)=1.\) Let \(n\) be a positive integer. Since \(D\) is of dimension one, the short exact sequence in the proof of Lemma 16 induces a long exact sequence
Thus, the first statement holds. Assume that \(n<m\) and \(F(n)=1.\) Then Lemma 16 gives the equality \(h^0(\fancyscript{O}_{nD})=1.\) Since the pull-back of any non-zero constant function on \((n+1)D\) via the canonical closed immersion \(nD\rightarrow (n+1)D\) is non-zero, the third homomorphism in the above long exact sequence is surjective. Thus, the equality
holds. Therefore, the second statement follows from Lemma 16. Since \(\nu \mid m,\) the last statement follows from Lemma 17. \(\square \)
Proposition 13
Let \(p\) be the characteristic of the residue field of \(\fancyscript{O}_{C,q}.\) Then there exists a non-negative integer \(e\) such that the equality \(m=\nu p^e\) holds. Here, we put \(p^0:=1.\)
Proof
For each positive integer \(n,\) let \(\nu _n\) be the order of the line bundle \(\fancyscript{O}_X(D)|_{nD}\) on \(nD\) in \(\text{ Pic}nD.\) Lemma 6.4.4 in [34] implies that there exists a positive integer \(n_0\) such that the equality \(\nu _n=m\) holds for any integer \(n\ge n_0.\) Let \(n\) be a positive integer. Let us denote the connecting homomorphism in the long exact sequence in the proof of Proposition 12 by \(\gamma _n.\) The short exact sequence of abelian sheaves on \(X\)
induces a long exact sequence
Thus, the relation \(\nu _n\mid \nu _{n+1}\) holds. Let us denote the connecting homomorphism in the above long exact sequence by \(\gamma _n^\times .\) The isomorphism
of abelian sheaves induces an isomorphism
of abelian groups. Since the equality \(\alpha _n(\text{ Im}\gamma _n)=\text{ Im}\gamma _n^\times \) holds (Proposition in [33, §6, p. 8]) and the abelian group \(H^1(X,\fancyscript{N}_n)\) is a \(p\)-torsion group, the quotient \(\nu _{n+1}/\nu _n\) is a power of \(p.\) Thus, the equalities \(\nu _1=\nu \) and \(\nu _{n_0}=m\) show the lemma. \(\square \)
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Mitsui, K. Logarithmic transformations of rigid analytic elliptic surfaces. Math. Ann. 355, 1123–1170 (2013). https://doi.org/10.1007/s00208-012-0813-1
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DOI: https://doi.org/10.1007/s00208-012-0813-1