Abel Maps and Limit Linear Series for Curves of Compact Type with Three Irreducible Components

Abstract

We explore the relationship between limit linear series and fibers of Abel maps for compact type curves with three components. For compact type curves with two components, given an exact Osserman limit linear series ${\mathfrak {g}}$, Esteves and Osserman associated a closed subscheme ${\mathbb {P}}({\mathfrak {g}})$ of the fiber of the corresponding Abel map. We generalize this definition to our case. Then, for ${\mathfrak {g}}$ the unique exact extension of an r-dimensional refined Eisenbud–Harris limit linear series, we find the irreducible components of ${\mathbb {P}}({\mathfrak {g}})$ and we show that ${\mathbb {P}}({\mathfrak {g}})$ is connected of pure dimension r, with the same Hilbert polynomial as the diagonal in ${\mathbb {P}}^{r}\times {\mathbb {P}}^{r}\times {\mathbb {P}}^{r}$.