Geometry of Families of Nodal Curves on the Blown{up Projective Plane

Let P2r be the projective plane blown up at r generic points. Denote by E0, E1, . . . , Er the strict transform of a generic straight line on P and the exceptional divisors of the blown–up points on P2r respectively. We consider the variety Virr(d; d1, . . . , dr ; k) of all irreducible curves C in |dE0 − ∑ r i=1 diEi| with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non–emptyness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ≤ 9 we give the complete answer concerning the existence of nodal curves in Virr(d; d1, . . . , dr ; k). Introduction We deal with the following general problem: given a smooth rational surface S and a divisor D on S, when is the variety Virr(D, k) of nodal irreducible curves in the complete linear system |D| with a fixed number k of nodes non–empty, when non– singular and when irreducible? For S = P, these questions are completely answered by the classical result of F. Severi ([Sev]), stating that the variety Virr(dH, k) of irreducible curves of degree d having k nodes is non–empty and smooth exactly if 0 ≤ k ≤ (d− 1)(d− 2) 2 , and the result of J. Harris ([Har]), stating that Virr(dH, k) is always irreducible. A modification of Severi’s method did lead to a sufficient (smoothness–)criterion for general smooth rational surfaces S ([Ta1, Nob]): let C0 ⊂ S be a smooth irreducible curve, let C ∈ |C0| be a reduced (nodal) curve with precisely k nodes, such that C = C1 ∪ . . . ∪ Cs, Ci irreducible and KS · Ci < 0 (0.0.1) for each 1 ≤ i ≤ s, then the variety Virr(C0, k) of irreducible curves in |C0| having precisely k nodes is smooth (see [Ta2, GrM, GrK, GrL] for generalizations to other surfaces). Moreover, in those cases each node of C can be smoothed independently. In this paper, we concentrate on the case S = Pr, the projective plane blown up at r generic points p1, . . . , pr. Let E0, E1, . . . , Er denote the strict transform of a generic straight line on P and the exceptional divisors of the blown–up points on P 2 r, respectively. Then for an irreducible nodal curve C ∈ |dE0 − ∑r i=1 diEi| the 1 2 G.-M. GREUEL, C. LOSSEN, AND E. SHUSTIN condition (0.0.1) reads as 3d > r