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Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...

Let a smooth projective curve $X$ over $\mathbb{C}$. Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$. Let $N=\deg (D)$ and $X^...