Let \(X_n\) be the projective plane blown up at \(n \geqslant 10\) general points. In this paper we give several consequences of the Segre–Harbourne–Gimigliano–Hirschowitz Conjecture, that pertain to complete linear systems on \(X_n\) . We begin by classifying such systems |C| with general irreducible member of genus \(g \geqslant 2\) (up to Cremona equivalence), in terms of invariants of the adjoint systems \(|C+mK|\) . We then use this to prove that, for fixed \(n \geqslant 10\) and \(g\geqslant 2\) , up to the action of the Cremona group, there exist finitely many complete linear systems on \(X_n\) whose general member is irreducible of genus g. Further, there is a function \(g\mapsto n(g)\) such that every such (effective) system is Cremona equivalent to a system in \(X_{n(g)}\) . The latter result is based on the explicit computation of the minimum possible self-intersection of an irreducible linear system with given n and \(\dim (|C|)\) . We classify those systems which achieve the minimal self-intersection. We also classify the systems with \(C^2 \leqslant 5\) , whether or not they have minimal \(C^2\) for the given n and dimension. We finish by proving several statements concerning systems that are base-point-free, and systems that give birational maps to their image.
