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We study the birational geometry of      M ¯  3 , 1 \overline {M}_{3,1}       and      M ¯  4 , 1 \overline {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.

Birational contractions of \overline{𝑀}_{3,1} and \overline{𝑀}_{4,1}