SMOOTHNESS AND IRREDUCIBILITY OF VARIETIES OF PLANE-CURVES WITH NODES AND CUSPS

Let V(d, m, k) be the variety of plane projective irreducible curves of degree d with m nodes and k cusps as their only singularities. We prove that V(d, m, k) is non-empty, non-singular and irreducible when m + 2k < alphad2, where alpha is some absolute explicit constant. This estimate is optimal with respect to the exponent of d