Geodesic
Classification of Algebraic Surfaces withSectional Genus less than or Equal to Six.II: Ruled Surfaces with dim φKx⊗L(x) = l
Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1110-1121

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Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].If g = 2 ≠ h l,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.Since X is ruled, h 2,0(X) = 0 and see [4] and [12, p. 390]. 
Livorni, Elvira Laura. Classification of Algebraic Surfaces withSectional Genus less than or Equal to Six.II: Ruled Surfaces with dim φKx⊗L(x) = l. Canadian journal of mathematics, Tome 38 (1986) no. 5, pp. 1110-1121. doi: 10.4153/CJM-1986-055-5
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