Abstract: 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 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 the Landau-Ginzburg/Calabi-Yau correspondence. In fact, we consider the moduli space of quasimaps of degree 1 to a point over the moduli stack of n pointed prestable 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 which asks whether the cohomology of the moduli space is a permutation representation or not. Based on a joint work with J. Choi and D.-K. Lee.
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