Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

Abstract

Abstract We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M2,2(𝓩).