On the Chow ring of the stack of rational nodal curves

From the introduction: In this thesis we start investigating the intersection theory of the Artin stack M0 of nodal curves of genus 0, following a suggestion of Rahul Pandharipande. Intersection theory on moduli spaces of stable curves has a not so long, but very intense history. It started at the beginning of the ’80’s with Mumford’s paper [Mum2], where he laid the foundations and carried out the first calculations. Many people have contributed to the theory after this (such as Witten and Kontsevich), building an imposing structure. The foundations of intersection theory on Deligne-Mumford stacks have been developed by Gillet and Vistoli. The first step towards an intersection theory on general Artin stacks (like those that arise from looking at unstable curves) was the equivariant intersection theory that Edidin and Graham developed, following an idea of Totaro. Their theory associates a commutative graded Chow ring A (M) with every smooth quotient stack M of finite type over a field. Unfortunately many stacks of geometric interest are not known to be quotient stacks (the general question of when a stack is a quotient stack is not well understood). Later, A. Kresch developed an intersection theory for general Artin stacks; in particular, he associates a Chow ring A (M) with every smooth Artin stack M locally of finite type over a field, provided some technical conditions hold, which are satisfied in particular for stacks of pointed nodal curves of fixed genus. Since there is not yet a theory of Chow rings of such stacks that extends the theory of stacks of stable curves, there does not seem to be much else to do than look at specific examples: and the first example is the stack M0 of nodal connected curves of genus 0. However, even this case turns out to be extremely complicated. In this thesis we compute the rational Chow ring of the open substack M 3 0 consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations. The techniques that we use, and the problems that we encounter, are discussed below.