Polarized surfaces of Δ-genus 3

Fania, Maria Lucia; Livorni, Elvira Laura

doi:10.1090/S0002-9947-1991-0992607-7

Polarized surfaces of $\Delta$-genus $3$
HTML articles powered by AMS MathViewer

by Maria Lucia Fania and Elvira Laura Livorni

Trans. Amer. Math. Soc. 328 (1991), 445-463

DOI: https://doi.org/10.1090/S0002-9947-1991-0992607-7

PDF | Request permission

Abstract:

Let $X$ be a smooth, complex, algebraic, projective surface and let $L$ be an ample line bundle on it. Let $\Delta = \Delta (X,L)= {c_1}{(L)^2} + 2 - {h^0}(L)$ denote the $\Delta$-genus of the pair $(X,L)$. The purpose of this paper is to classify such pairs under the assumption that $\Delta = 3$ and the complete linear system $| L |$ contains a smooth curve. If $d \geq 7$ and $g \geq \Delta$, Fujita has shown that $L$ is very ample and $g= \Delta$. If $d \geq 7$ and $g < \Delta = 3$, then $g= 2$ and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of $L + tK$ being nef or not, for $t= 1,2$. In the cases in which $L + 2K$ is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If $g > \Delta$ there are still open cases to solve in which completely different methods are needed.

References

Mauro Beltrametti, Antonio Lanteri, and Marino Palleschi, Algebraic surfaces containing an ample divisor of arithmetic genus two, Ark. Mat. 25 (1987), no. 2, 189–210. MR 923406, DOI 10.1007/BF02384443
Aldo Biancofiore and Elvira Laura Livorni, On the iteration of the adjunction process in the study of rational surfaces, Indiana Univ. Math. J. 36 (1987), no. 1, 167–188. MR 876997, DOI 10.1512/iumj.1987.36.36009
Aldo Biancofiore and Elvira Laura Livorni, On the iteration of the adjunction process for surfaces of negative Kodaira dimension, Manuscripta Math. 64 (1989), no. 1, 35–54. MR 994380, DOI 10.1007/BF01182084
Aldo Biancofiore and Elvira Laura Livorni, Algebraic ruled surfaces with low sectional genus, Ricerche Mat. 36 (1987), no. 1, 17–32. MR 977050
Takao Fujita, On the structure of polarized varieties with $\Delta$-genera zero, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103–115. MR 369363
Takao Fujita, On polarized varieties of small $\Delta$-genera, Tohoku Math. J. (2) 34 (1982), no. 3, 319–341. MR 676113, DOI 10.2748/tmj/1178229197
Takao Fujita, On hyperelliptic polarized varieties, Tohoku Math. J. (2) 35 (1983), no. 1, 1–44. MR 695657, DOI 10.2748/tmj/1178229099
Takao Fujita, On polarized manifolds of $\Delta$-genus two. I, J. Math. Soc. Japan 36 (1984), no. 4, 709–730. MR 759426, DOI 10.2969/jmsj/03640709
Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 167–178. MR 946238, DOI 10.2969/aspm/01010167

Classification of polarized manifolds of sectional genus two

On classification of polarized manifolds by sectional genus

Takao Fujita, Polarized manifolds of degree three and $\Delta$-genus two, J. Math. Soc. Japan 41 (1989), no. 2, 311–331. MR 984755, DOI 10.2969/jmsj/04120311
Takao Fujita, On the structure of polarized manifolds with total deficiency one. III, J. Math. Soc. Japan 36 (1984), no. 1, 75–89. MR 723595, DOI 10.2969/jmsj/03610075
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Paltin Ionescu, Ample and very ample divisors on surfaces, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 4, 349–358. MR 950131
Antonio Lanteri, On polarized surfaces of $\Delta$-genus two, Ann. Mat. Pura Appl. (4) 151 (1988), 317–329 (English, with Italian summary). MR 964517, DOI 10.1007/BF01762802
Antonio Lanteri and Marino Palleschi, About the adjunction process for polarized algebraic surfaces, J. Reine Angew. Math. 352 (1984), 15–23. MR 758692
Elvira Laura Livorni, Classification of algebraic surfaces with sectional genus less than or equal to six. III. Ruled surfaces with $\textrm {dim}\phi _{K_X\otimes L}(X)=2$, Math. Scand. 59 (1986), no. 1, 9–29. MR 873485, DOI 10.7146/math.scand.a-12150
Elvira Laura Livorni, Classification of algebraic surfaces with sectional genus less than or equal to six. II. Ruled surfaces with $\textrm {dim}\,\phi _{K_X\otimes L}(X)=1$, Canad. J. Math. 38 (1986), no. 5, 1110–1121. MR 869716, DOI 10.4153/CJM-1986-055-5
Andrew John Sommese, Hyperplane sections of projective surfaces. I. The adjunction mapping, Duke Math. J. 46 (1979), no. 2, 377–401. MR 534057

On the birational theory of hyperplane sections of projective threefolds

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