On the second adjunction mapping. The case of a 1-dimensional image

Beltrametti, Mauro; Sommese, Andrew

doi:10.1090/S0002-9947-97-01809-6

On the second adjunction mapping. The case of a $1$-dimensional image
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by Mauro C. Beltrametti and Andrew J. Sommese

Trans. Amer. Math. Soc. 349 (1997), 3277-3302

DOI: https://doi.org/10.1090/S0002-9947-97-01809-6

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Abstract:

Let $\widehat {L}$ be a very ample line bundle on an $n$-dimensional projective manifold $\widehat {X}$, i.e., assume that $\widehat {L}\approx i^*\mathcal {O}_{\mathbb {P}_ N}(1)$ for some embedding $i:\widehat {X}\hookrightarrow \mathbb {P}_ N$. In this article, a study is made of the meromorphic map, $\widehat {\varphi } : \widehat {X}\to \Sigma$, associated to $|K_{\widehat {X}}+(n-2)\widehat {L}|$ in the case when the Kodaira dimension of $K_{\widehat {X}}+(n-2)\widehat {L}$ is $\ge 3$, and $\widehat {\varphi }$ has a $1$-dimensional image. Assume for simplicity that $n=3$. The first main result of the paper shows that $\widehat \varphi$ is a morphism if either $h^0(K_{\widehat X}+\widehat L)\geq 7$ or $\kappa (\widehat {X})\geq 0$. The second main result of this paper shows that if $\kappa (\widehat X)\ge 0$, then the genus, $g(f)$, of a fiber, $f$, of the map induced by $\widehat \varphi$ on hyperplane sections is $\leq 6$. Moreover, if $h^0(K_{\widehat X}+\widehat L)\ge 21$ then $g(f)\leq 5$, a connected component $F$ of a general fiber of $\widehat \varphi$ is either a $K3$ surface or the blowing up at one point of a $K3$ surface, and $h^1(\mathcal {O}_{\widehat X})\le 1$. Finally the structure of the finite to one part of the Remmert-Stein factorization of $\widehat \varphi$ is worked out.

References

W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
Arnaud Beauville, L’application canonique pour les surfaces de type général, Invent. Math. 55 (1979), no. 2, 121–140 (French). MR 553705, DOI 10.1007/BF01390086
Mauro Beltrametti, Michael Schneider, and Andrew J. Sommese, Threefolds of degree $11$ in $\textbf {P}^5$, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 59–80. MR 1201375, DOI 10.1017/CBO9780511662652.006
Mauro C. Beltrametti, Michael Schneider, and Andrew J. Sommese, Some special properties of the adjunction theory for $3$-folds in $\textbf {P}^5$, Mem. Amer. Math. Soc. 116 (1995), no. 554, viii+63. MR 1257080, DOI 10.1090/memo/0554
M. C. Beltrametti and A. J. Sommese, On the dimension of the adjoint linear system for threefolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 1–24. MR 1315348
Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR 1318687, DOI 10.1515/9783110871746
Takao Fujita, Theorems of Bertini type for certain types of polarized manifolds, J. Math. Soc. Japan 34 (1982), no. 4, 709–718. MR 669278, DOI 10.2969/jmsj/03440709
Joe Harris, Curves in projective space, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 85, Presses de l’Université de Montréal, Montreal, Que., 1982. With the collaboration of David Eisenbud. MR 685427
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Paltin Ionescu, Embedded projective varieties of small invariants, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR 749942, DOI 10.1007/BFb0071773
Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
A. Lanteri, M. Palleschi, and A. J. Sommese, On triple covers of $\mathbf P^n$ as very ample divisors, Classification of algebraic varieties (L’Aquila, 1992) Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 277–292. MR 1272704, DOI 10.1090/conm/162/01537
Elvira Laura Livorni, Classification of algebraic nonruled surfaces with sectional genus less than or equal to six, Nagoya Math. J. 100 (1985), 1–9. MR 818155, DOI 10.1017/S0027763000000192
Andrew John Sommese, Hyperplane sections of projective surfaces. I. The adjunction mapping, Duke Math. J. 46 (1979), no. 2, 377–401. MR 534057
Andrew John Sommese, Hyperplane sections, Algebraic geometry (Chicago, Ill., 1980) Lecture Notes in Math., vol. 862, Springer, Berlin-New York, 1981, pp. 232–271. MR 644822
Andrew John Sommese, On the minimality of hyperplane sections of projective threefolds, J. Reine Angew. Math. 329 (1981), 16–41. MR 636441, DOI 10.1515/crll.1981.329.16
Andrew John Sommese, On the adjunction theoretic structure of projective varieties, Complex analysis and algebraic geometry (Göttingen, 1985) Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 175–213. MR 855885, DOI 10.1007/BFb0077004
Andrew J. Sommese, On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold, J. Reine Angew. Math. 402 (1989), 211–220. MR 1022801, DOI 10.1515/crll.1989.402.211
Hajime Tsuji, Stability of tangent bundles of minimal algebraic varieties, Topology 27 (1988), no. 4, 429–442. MR 976585, DOI 10.1016/0040-9383(88)90022-5
E. Viehweg, Quasi-Projective Moduli for Polarized Manifolds, Ergeb. Math. Grenzgeb. (3) 30 (1995), Springer, Berlin.