• K-theory
• algebraic stacks
• genus 2 curves
• equivariant geometry

The stack of smooth genus 2 curves, denoted M_2, is an object which parametrizes all families of smooth genus 2 curves. It has a natural, intrinsic geometry (it is a smooth Deligne-Mumford stack) which captures properties of families of curves. So it is interesting to compute various geometric invariants of M_2.

The goal here is to compute its Grothendieck ring, K_0(M_2). The Grothendieck ring of a scheme or a stack is a subtle invariant of its category of coherent (or locally free) sheaves. For smooth schemes it is closely related to the Chow group (i.e. algebraic cycles modulo rational equivalence), as the two become isomorphic upon tensorisation with Q via the Chern character. For stacks of curves this is false because the integral Chow group is almost all torsion - so the link between the two groups is less evident.

Fortunately for us they share so many formal properties (excision, a projective bundle formula, a self-intersection formula etc.) that it is possible to mimic what has been done for the Chow group to compute K_0. Thus we follow closely the approach Vistoli takes in the article where he computes the Chow ring of M_2. The first step is a description of M_2's geometry which makes computations actually possible - and translates the problem into one of equivariant geometry. A few reduction steps follow, and we are left to compute the push forward of some classes. Here we diverge from Vistoli's article and use a localisation theorem for equivariant K-theory to finish off the computation.