Let $S\subset\mathbb{P}^g$ be a smooth polarized K3 surface of genus $g$. I am interested in the existence of certain cuspidal curves in the linear system. We know a general hyperplane section $H\cap S$ is smooth, and to have a nodal singularity is a codimension 1 condition (Let's say the dual variety of $S$ is a divisor in $(\mathbb{P}^g)^{\lor}$). However, I am interested in the question of having a hyperplane section of multiple cuspidal singularities. The main case is when $g=8$ and I need a hyperplane of at least $3$ cusps. I could not find a proof or disproof for myself. Any input is very helpful. Thanks!

Let $S\subset\mathbb{P}^g$ be a smooth polarized K3 surface of genus $g$. I am interested in the existence of certain cuspidal curves in the linear system. We know a general hyperplane section $H\cap S$ is smooth, and to have a nodal singularity is a codimension 1 condition (Let's say the dual variety of $S$ is a divisor in $(\mathbb{P}^g)^{\lor}$). However, I am interested in the question of having a hyperplane section of multiple cuspidal singularities. The main case is when $g=8$ and I need a hyperplane of at least $3$ cusps. I could not find a proof or disproof for myself. Any input is very helpful. Thanks!

I guess I should add that I only want the existence for some K3 surfaces.