A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

We construct a modular desingularisation of (M) over bar (2,n)(P-r, d)(main). The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers; with this enhanced logarithmic structure, it is possible to desingularise the main component by means of a logarithmic modification. Both isolated and nonreduced singularities appear naturally. Our construction gives rise to a notion of reduced Gromov-Witten invariants in genus two.