On elements of degree $g$ in the separating semigroup of a real curve of genus $g$

Source abstract

Let $C$ be a separating real algebraic curve of genus $g$ whose real locus consists of $r$ components and let $S=\mathrm{Sep}(C)$ be its separating semigroup. An element $δ\in S$ is called extendable if $δ+\mathbb N_0^r\subset S$, where $\mathbb N_0=\{n\in\mathbb Z\mid n\ge 0\}$. We prove that an element of $S$ is extendable if it is realizable by a separating mobile divisor of degree $g$ and projective rank 1.