Intersection theory on moduli spaces of curves (Q2707379)

Aim of this book is to introduce the reader to the intersection theory on the moduli space of curves, starting from the basic definitions of intersection theory and arriving at the most recent conjectures on the tautological ring of \(M_g\). NEWLINENEWLINENEWLINEOriginally written as lecture notes of a course, the book is mainly addressed to graduate students already having a good knowledge of schemes and sheaf cohomology. The first five chapters contain the definitions and preliminary results, not all elementary, which are needed to make the book self-contained: fine and coarse moduli spaces, the coarse moduli space \(M_g\) of smooth projective curves of genus \(g\), its Deligne-Mumford compactification \(\overline M_g\) constructed via Kuranishi families, the Kontsevich moduli spaces of stable maps, the Chow rings of smooth varieties and of varieties with ``nice'' singularities, the jets extension of relative bundles. Chapters 6 and 7, which form the core of the book, contain the main results of intersection theory on \(M_g\) and \(\overline M_g\). The theory is illustrated by some fine examples of computation of classes in the Chow ring of \(M_g\) and \(\overline M_g\): Some naturally defined loci are considered, as the hyperelliptic locus in \(M_g\) or the closure in \(\overline M_g\) of \(wt(2)\), the locus of curves having a Weierstrass point of weight \(\geq 2\). Finally, chapter 8 is devoted to explain the Carel Faber conjectures on the tautological ring of \(M_g\). As a matter of example of the techniques introduced in the previous sections, the tautological rings of curves of genus 3 and 4 are computed. NEWLINENEWLINENEWLINEEven though the book does not give proofs of most results, it is an effective introduction to these important topics, also thanks to the examples, which are numerous and developed in all details, and to some useful exercises.