HYPERTREES PROJECTIONS AND MODULI OF STABLE RATIONAL CURVES

HYPERTREES, PROJECTIONS, AND MODULI OF STABLE RATIONAL CURVES ANA-MARIA CASTRAVET AND JENIA TEVELEV ABSTRACT. We give a conjectural description for the cone of effective divisors of the Grothendieck–Knudsen moduli space M0,n of stable ra- tional curves with n marked points. Namely, we introduce new combi- natorial structures called hypertrees and show that they give exceptional divisors on M0,n with many remarkable properties. 1. INTRODUCTION A major open problem inspired by the pioneering work of Harris and Mumford [HM] on the Kodaira dimension of the moduli space of stable curves, is to understand geometry of its birational models, and in particular to describe its cone of effective divisors and a decomposition of this cone into Mori chambers [HK] encoding ample divisors on birational models. Here we study the genus zero case. The moduli spaces M0,n parame- trize stable rational curves, i.e., nodal trees of P1's with n marked points and without automorphisms. For any subset I of marked points, M0,n has a natural boundary divisor ?I whose general element parametrizes stable rational curves with two irreducible components, one marked by points in I and another marked by points in Ic. We will introduce new combinatorial objects called hypertrees with an eye towards the following 1.1. CONJECTURE. The effective cone of M0,n is generated by boundary divisors and by divisors D? (defined below) parametrized by irreducible hypertrees.

• ?? ?


• rays can


• hypertrees


• stable hypertree


• hypertree


• curve ??


• ?j


• triangulation has