×

On polarized surfaces \((X, L)\) with \(h^ 0 (L)> 0\), \(\kappa (X)= 2\), and \(g(L)= q(X)\). (English) Zbl 0878.14005

Let \(X\) be a smooth complex projective surface of irregularity \(q(X)\) endowed with a nef and big line bundle \(L\) of arithmetic genus \(g(L)\). It was conjectured by Fujita that \(g(L)\geq q(X)\) for every such a pair \((X,L)\). This conjecture is known to be true in some special cases (e.g. if \(L\) is effective, or if \(X\) has Kodaira dimension \(\kappa(X) \leq 1)\). When the conjecture is true it is natural to study for which pairs \((X,L)\) the equality \(g(L) =q(X)\) occurs. This investigation, started by the reviewer when \(L\) is ample [A. Lanteri, Boll. Un. Mat. Ital., VI. Ser., A 4, 391-400 (1985; Zbl 0597.14031)], was recently carried out by the author [Y. Fukuma, “A lower bound for the sectional genus of quasi-polarized surfaces”, Geom. Dedicata 64, No. 2, 229-251 (1997)]. Even assuming that \(L\) is ample, the case when \(X\) is of general type is the most delicate one to be studied. In the paper under review, the author proves that if \(\kappa (X)=2\) and \(L\) is ample and effective, then the equality \(g(L)= q(X)\) implies \(h^0(L) =1\) and \(1\leq L^2\leq 4\); moreover the effective divisor \(D\) in \(|L|\) is reduced and either it is an irreducible smooth curve, or \(X\) is a product and \(D\) is the sum of the two factors.
Reviewer: A.Lanteri (Milano)

MSC:

14C20 Divisors, linear systems, invertible sheaves
14J29 Surfaces of general type

Citations:

Zbl 0597.14031
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Beauville, L’inegalite \(p_{g}\geq 2q-4\) pour les surfaces de type général, Bull. Soc. Math. France 110 (1982), 343–346.
[2] E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171 – 219. · Zbl 0259.14005
[3] F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), no. 2, 483 – 515. · Zbl 0549.14012
[4] J.-P.Demailly, Effective bounds for very ample line bundles, Invent Math. 124 (1996), 243–261. CMP 96:06
[5] Takao Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. · Zbl 0743.14004
[6] Takao Fujita, On the structure of polarized varieties with \Delta -genera zero, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103 – 115. · Zbl 0333.14004
[7] Y. Fukuma, A lower bound for the sectional genus of quasi-polarized surfaces, to appear in Geometriae Dedicata. · Zbl 1049.14026
[8] ——, A lower bound for sectional genus of quasi-polarized manifolds, to appear in J. Math. Soc. Japan. · Zbl 0899.14003
[9] Antonio Lanteri, Algebraic surfaces containing a smooth curve of genus \?(\?) as an ample divisor, Geom. Dedicata 17 (1984), no. 2, 189 – 197. · Zbl 0548.14002 · doi:10.1007/BF00151507
[10] R. Lazarsfeld, Lectures on Linear Series, preprint. · Zbl 0906.14002
[11] Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269 – 331. · Zbl 1103.14301
[12] C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. (N.S.) 36 (1972), 41 – 51. · Zbl 0276.32018
[13] Igor Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. (2) 127 (1988), no. 2, 309 – 316. · Zbl 0663.14010 · doi:10.2307/2007055
[14] Fumio Sakai, Weil divisors on normal surfaces, Duke Math. J. 51 (1984), no. 4, 877 – 887. · Zbl 0602.14006 · doi:10.1215/S0012-7094-84-05138-X
[15] 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. · Zbl 0601.14029 · doi:10.1007/BFb0077004
[16] Andrew John Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann. 278 (1987), no. 1-4, 593 – 603. · Zbl 0655.14001 · doi:10.1007/BF01458083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.