Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

Let $\Gamma$ be a plane curve of degree $d$ with $\delta$ ordinary nodes and no other singularities. If $P$ is a smooth point on $\Gamma$ then the Weierstrass gap sequence at $P$ is considered as that at the corresponding point on the normalization of $\Gamma$. A smooth point $P\in\Gamma$ is called a total inflection point if $i(\Gamma ,T;P)=d$ where $T$ is the tangent line to $\Gamma$ at $P$. There are many possible Weierstrass gap sequences at total inflection points. Our main results are: Among them (1) There exists a pair $(P,\Gamma )$ such that the gap sequence at $P$ is the minimal (in the sense of weight). (2) There exists a pair $(P,\Gamma )$ such that the gap sequence at $P$ is the maximal (resp. up to 1 maximal). And we characterize these cases in the sense of location of nodes.