Let $C$ be a connected compact complex curve on a non-singular complex surface. We investigate function-theoretic properties of the neighborhood of $C$ by dividing the situation into three cases according to the sign of the largest eigenvalue $\lambda$ of its intersection matrix. Our contribution in the this paper is in the case $\lambda>0$. In this case, for a neighborhood $V$ of $C$, we show the existence of a strictly plurisubharmonic function on $V\setminus C$ which has a logarithmic growth along $C$.