Marc Coppens — Page 2

arXiv1993

Non-trivial Linear Systems on Smooth Plane Curves

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and $r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let $g^r_n$ be a base point free very special non-trivial complete linear system on $C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b \le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is best possible.

alg-geom↗

arXiv1992

Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

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

alg-geom↗