Geometry of families of nodal curves on the blown-up projective plane

Let ℙ²_r, be the projective plane blown up at r generic points. Denote by E₀,E₁,... ,E_r the strict transform of a generic straight line on ℙ² and the exceptional divisors of the blown-up points on ℙ²_r respectively. We consider the variety V_irr(d_i d₁,... , d_r; k) of all irreducible curves C in |dE₀ -∑_i=1r=d_iE_i| with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For r ≤ 9 we give the complete answer concerning the existence of nodal curves in V_irr(d; d₁,... ,d_r; k).