Cycles in the moduli spaces of stable curves

Abstract

In this chapter we study the Chow ring of moduli stacks of stable pointed curves. We first discuss the theory of cycles on stack quotients of smooth varieties by a finite group. We then introduce the κ classes, the Hodge classes, the point classes, and the boundary classes. Following Mumford, we establish relations among these classes via the flatness of the Gauss–Manin connection and via the Grothendieck–Riemann–Roch theorem. We then discuss the tautological ring of moduli stacks of stable pointed curves. Finally, we describe the Chow ring of the moduli spaces of genus zero curves.