GEOMETRY OF FAMILIES OF NODAL CURVES ON THE BLOWN-UP PROJECTIVE PLANE

Let P-r(2) be the projective plane blown up at r generic points. Denot e by E-0, E-1,..., E-r the strict transform of a generic straight line on P-2 and the exceptional divisors of the blown-up points on P-r(2) respectively. We consider the variety V-irr(d; d(1),...,d(r);k) of all irreducible curves C in \dE(0)- Sigma(i=1)(r)d(i)E(i) with k nodes as the only singularities and give asymptotically nearly optimal suffici ent conditions for its smoothness, irreducibility and non-emptiness. M oreover, we extend our conditions for the smoothness and the irreducib ility to families of reducible curves. For r equal to or less than 9 w e give the complete answer concerning the existence of nodal curves in V-irr(d; d(1),...,d(r); k).