Youngook Choi
Yeungnam University, Gyeongsan, Republic of Korea

Let $\mathcal M_g$ be the moduli space of smooth complex curves of genus $g$. It is well known that a general curve of genus $g$ has no linear series $g^r_d$ whose Brill-Noether number $\rho(g,r,d)(:=g-(r+1)(g-d+r))$ is negative. The Brill-Noether locus $\mathcal M^r_{g,d}$ is defined by the sublocus of $\mathcal M_g$ whose elements represent curves possessing a linear series $g^r_d$.

In 1987, D. Eisenbud and J. Harris proved that $\mathcal M_{23}$ has Kodaira dimension $\ge 1$ by showing that $\mathcal M^1_{23,12}\neq \mathcal M^2_{23,17}$. In 2000, G. Farkas established that $\mathcal M_{23}$ has Kodaira dimension $\ge 2$ by demonstrating that $\mathcal M^1_{23,12}$, $\mathcal M^2_{23,17}$, and $\mathcal M^3_{23,20}$ are mutually distinct.

In this talk, we discuss conditions for the existence/non-existence of a smoothable limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. This work gives relations among Brill-Noether loci of codimension at most two in the moduli space of complex curves and shows Brill-Noether loci of codimension two have mutually distinct supports.