Extremal values of the basic invariants of plane curves

In [A2] V.I. Arnold introduced three basic invariants St, J(+) and J(-) of plane curves and proposed some interesting conjectures concerning the extre mal values of these invariants on a given set of curves. Partial answers ha ve been obtained by O. Viro and A.N. Shumakovich. We give explicit formulas for these extremal values of sets of plane curves with fixed number of dou ble points and of Whitney index and we determine on which curves these extr emal values are attained (Theorems 3-6). Our arguments are based on underst anding of the fine structures of generic curves and some surgery operations on curves.