Let S be a compact complex surface of general type, Smin its minimal model gotten by contracting the finite number of (−1)-curves. It is a smooth surface with nef canonical line bundle KSmin . It then can be shown that Smin has at most b2 curves on which the canonical line bundleKSmin restricts to a non-ample bundle (i.e. whose intersection number with KSmin is non positive). These are (−2) smo...

In 1932 F. Severi claimed, with an incorrect proof, that every smooth minimal pro-jective surface S such that the bundle Ω 1 S is generically generated by global sections satisfies the topological inequality 2c 2 1 (S) ≥ c2(S). According to Enriques-Kodaira classification, the above inequality is easily verified when the Kodaira dimension of the surface is ≤ 1, while for surfaces of general typ...

Smooth projective varieties with small invariants have got renewed interest in recent years, primarily due to the fine study of the adjunction mapping by Reider, Sommese, Van de Ven and others. For the special case of smooth surfaces in P the method goes back to the Italian geometers, who at the turn of the century used it for the study of the surfaces of degree less than 7, or sectional genus ...

Given a minimal surface S equipped with a generically finite map to an Abelian variety and C ⊂ S a rational or an elliptic curve, we show that the canonical degree of C is bounded by four times the self-intersection of the canonical divisor of S. As a corollary, we obtain the finiteness of rational and elliptic curves with an optimal uniform bound on their canonical degrees on any surface of ge...

We give many examples of surfaces of general type with pg = 0 for which Bloch’s conjecture holds, for all values of K2 6= 9. Our surfaces are equipped with an involution. Let S be a smooth complex projective surface with pg(S) = 0. Bloch’s conjecture states that the Albanese map A0(S)0 → Alb(S) is an isomorphism, where A0(S)0 is the Chow group of 0-cycles of degree 0 on S. It is known for all s...

Smooth projective varieties with small invariants have got renewed interest in recent years, primarily due to the ne study of the adjunction mapping by Reider, Sommese, Van de Ven and others. For the special case of smooth surfaces in P 4 the method goes back to the Italian geometers, who at the turn of the century used it for the study of the surfaces of degree less than 7, or sectional genus ...