Birational Contractions of ${\bar{M}}_{3,1}$ and ${\bar{M}}_{4,1}$

Abstract

We study the birational geometry of ${\bar{M}}_{3,1}$ and ${\bar{M}}_{4,1}$. In particular, we pose a pointed analogue of the Slope Conjecture and prove it in these low-genus cases. Using variation of GIT, we construct birational contractions of these spaces in which certain divisors of interest -- the pointed Brill-Noether divisors -- are contracted. As a consequence, we see that these pointed Brill-Noether divisors generate extremal rays of the effective cones for these spaces.