 
Summary: WEAKLY PROPER MODULI STACKS OF CURVES
JAROD ALPER, DAVID SMYTH, AND FREDERICK VAN DER WYCK
ABSTRACT. This is the first in a projected series of three papers in which we construct
the second flip in the log minimal model program for Mg. We introduce the notion of
a weakly proper algebraic stack, which may be considered as an abstract character
ization of those mildly nonseparated moduli problems encountered in the context of
Geometric Invariant Theory (GIT), and develop techniques for proving that a stack is
weakly proper without the usual semistability analysis of GIT. We define a sequence of
moduli stacks of curves involving nodes, cusps, tacnodes, and ramphoid cusps, and use
the aforementioned techniques to show that these stacks are weakly proper. This will
be the key ingredient in forthcoming work, in which we will prove that these moduli
stacks have projective good moduli spaces which are log canonical models for Mg.
CONTENTS
1. Introduction 1
2. Weak properness 8
3. Sm,1/Sm,2curves and Hm,1/Hm,2curves 14
4. A
k /Ak/A+
k stability 18
5. Closed points of Mg,n(Ak) 25
