We prove that there exists a positive integer νn depending only on n such that for every smooth projective n-fold of general typeX defined over complex numbers, | mKX | gives a birational rational map from X into a projective space for every m ≥ νn. This theorem gives an affirmative answer to Severi’s conjecture.

Abstract Let $(S,L)$ be a general polarised Enriques surface, with L not numerically 2-divisible. We prove the existence of regular components all Severi varieties irreducible nodal curves in linear system $|L|$ , that is, for any number nodes $\delta =0, \ldots p_a(L)-1$ . This solves classical open problem and gives positive answer to recent conjecture Pandharipande–Schmitt, under additional ...