Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. We give complete classification surfaces for which equality holds $q(X)\geq 3$: these are whose canonical model is double cover product elliptic branched over an ample divisor at most negligible singularities intersects the fibre twice...

Continuing the work of Chiantini and Ciliberto on the Severi varieties of curves on surfaces in P, we complete the proof of the existence of regular components for such varieties. 2000 Mathematics Subject Classification. Primary 14H10, 14B07

We classify all Jacobian elliptic fibrations on K3 surfaces with finite automorphism group. also extremal infinite group and 2-elementary N\'eron-Severi lattice.