Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

C be the normalization of Γ . Let g = (d− 1)(d− 2) 2 − δ; the genus of C. We identify smooth points of Γ with the corresponding points on C. In particular, if P is a smooth point on Γ then the Weierstrass gap sequence at P is considered with respect to C. A smooth point P ∈ Γ is called an (e − 2)-inflection point if i(Γ, T ;P ) = e ≥ 3 where T is the tangent line to Γ at P (cf. Brieskorn–Knörrer[1, p. 372]). Of course, e ≤ d and a 1-inflection point is an ordinary flex. In particular, a (d − 2)-inflection point is called a total inflection point. Let N be the semigroup consisting of the non-gaps of P , so N − N = {α1 < α2 < · · · < αg} is the Weierstrass gap sequence of P . Clearly {d − 1, d} ⊂ N , so Nd := {a(d− 1) + bd|a, b ∈ N} ⊂ N (see also Lemma 1.2).