A modular smooth compactification of genus 2 curves in projective spaces
Moduli spaces of stable maps in genus bigger than zero include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe modularly.
On the other hand, after the work of Li-Vakil-Zinger and Ranganathan-Santos-Parker-Wise in genus one, we know that points in the boundary of the main component correspond to maps that admit a nice factorisation through some curve with Gorenstein singularities on which the map is less degenerate. Morally, such a Gorenstein curve is obtained by contracting any higher genus sub-curve on which the map is constant.
The question becomes how to construct such a universal family of Gorenstein curves to then single out the (resolution) of the main component of maps imposing the factorization property. In joint work with L. Battistella, we construct one such family in genus two over a logarithmic modification of the space of admissible covers, and consequently obtain the desired smooth compactification of genus 2 curves in projective spaces.