Cohen-Macaulayness of quiver Grassmannians arising from limit linear series.

Linear series on families of smooth curves degenerating to a nodal curve $X$ give rise to special quivers $Q$, called $\mathbb{Z}^n$-quivers, where $n+1$ is the number of components of $X$, and certain special representations $\mathfrak{g}$ of those quivers, called linked nets. The associated quiver Grassmannians to each $\mathfrak{g}$ have been studied by Osserman for $n=1$. Here we study that parameterizing subrepresentations of pure dimension 1, called the linked projective space $\mathbb{LP}(\mathfrak{g})$, for higher $n$. In particular, for $n=2$, we describe the points of $\mathbb{LP}(\mathfrak{g})$, its nonsingular locus and show that $\mathbb{LP}(\mathfrak{g})$ is Cohen-Macaulay, reduced and of pure dimension $\dim(\mathfrak{g})-1$ if $\mathfrak{g}$ is exact.