Young-Hoon Kiem, A New Construction of the Moduli Space of Pointed Stable Curves of Genus 0

The moduli space of n points on a projective line up to projective equivalence has been a topic of research since the 19th century. A natural moduli theoretic compactification was constructed by Deligne and Mumford as an algebraic stack. Later, Knudsen, Keel, Kapranov and others provided explicit constructions by sequences of blowups. The known inductive constructions of Knudsen and Keel however are rather inconvenient when one wants to compute the cohomology of the compactified moduli space as a representation space of its automorphism group because the blowup sequences are not equivariant. I will talk about a new inductive construction of the much studied moduli space from the perspective of sheaf theory. In fact, we consider the moduli space of rank 1 stable pairs over the moduli space of n pointed stable curves of genus 0. By studing the wall crossing, we obtain an equivariant sequence of blowups which ends up with the moduli space of n+1 pointed stable curves of genus 0. As an application, we provide a closed formula of the character of the cohomology of the moduli space. We also provide a partial answer to a question of Manin and Orlov which asks whether the cohomology is a permutation representation or not. Based on a joint work with Jinwon Choi and Donggun Lee.