Resumen de Moduli of relative stable maps to P1: cut-and-paste invariants

Siddarth Kannan

• We study constructible invariants of the moduli space M(x) of stable maps from genus zero curves to P1, relative to 0 and ∞, with ramification profiles specified by x ∈ Zn. These spaces are central to the enumerative geometry of P1, and provide a large family of birational models of the Deligne–Mumford–Knudsen moduli space M0,n. For the sequence of vectors x corresponding to maps which are maximally ramified over 0 and unramified over ∞, we prove that a generating function for the topological Euler characteristics of these spaces satisfies a differential equation which allows for its recursive calculation. We also show that the class [M(x)] ∈ K0(Var/C) of the moduli space in the Grothendieck ring of varieties is constant as x varies within a fixed chamber in the resonance decomposition of Zn. We conclude by suggesting several further directions in the study of these spaces, giving conjectures on (1) the asymptotic behavior of the Euler characteristic and (2) a potential chamber structure for the Chern numbers.