On the First Steps of the Minimal Model Program for the Moduli Space of Stable Pointed Curves

Abstract

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

On some modular contractions of the moduli space of stable pointed curves

“…2.1]. Since the number of these birational contractions grows exponentially in (g, n) (see Remark 3.5 for the exact count), this significantly expands the known examples of birational contractions of M g,n , that previous to [CTV18], to the best of our knowledge, consisted of the first two steps of the Hassett-Keel program (see [HH13], [AFSvdW17,AFS17b,AFS17a]) and, for n = 0, the Torelli morphism from M g to the Satake compactification of the moduli space of principally polarized abelian varieties.…”

“…In our previous paper [CTV18], we introduced several new birational contractions Υ T : M g,n → M T g,n , whose codomains M T g,n are weakly modular compactifications of M g,n in the sense of [FS13, Sec. 2.1].…”

“…The aim of this paper, which is a sequel of [CTV18], is to study the geometry of the variety M T g,n and of the birational contraction Υ T : M g,n → M T g,n . Moreover, we describe Υ T : M g,n → M T g,n as the ample model of suitable adjoint divisors on M g,n (see Theorem 1.1).…”

“…Let us also mention that in [CTV18] we also constructed several other weakly modular compactifications M T + g,n of M g,n , which are endowed with a morphism f + T : M T + g,n → M T g,n that is a flip (with respect to a suitable divisor) of Υ T : M g,n → M T g,n . However, the varieties M T + g,n are only rational (and not regular) contractions of M g,n and we will not study them in this paper.…”

“…As special case of the above stacks, for T = ∅ we obtain the stack of pseudostable n-pointed curves of genus g M ps g,n := M ∅ g,n = {n-pointed curves of genus g with ample log canonical class, having singularities that are nodes and cusps, and not having elliptic tails}. Excluding the trivial case (g, n) = (1, 1) (when M ps g,n = ∅) and the pathological case (g, n) = (2, 0) (see [CTV18,Rmk. 1.13] and also Remark 3.8), M ps g,n is a proper Artin stack with finite inertia (and Deligne-Mumford if char(k) = 2, 3) with coarse moduli space φ ps : M ps g,n → M ps g,n which is a normal projective variety.…”