Journal of the
Korean Mathematical Society JKMS

Let $X$ be an integral Gorenstein projective curve with $g:= p_a(X)\geq 3$. Call $G^r_d(X,**)$ the set of all pairs $(L,V)$ with $L \in \text{Pic}(X),\ \text{\deg}(L) = d,\ V\subseteq H^0(X,L),\ \dim(V) = r+1$ and $V$ spanning $L$. Assume the existence of integers $d$, $r$ with $1\leq r\leq d\leq g-1$ such that there exists an irreducible component, $\Gamma$, of $G^r_d(X,**)$ with $\dim (\Gamma)\geq d-2r$ and such that the general $L\in \Gamma$ is spanned at every point of Sing$(X)$. Here we prove that $\dim(\Gamma) = d-2r$ and $X$ is hyperelliptic.

Keywords: Brill - Noether theory, singular projective curve, torsion free sheaf

MSC numbers: 14H51, 14H20