Stable maps spaces arise naturally when trying to answer questions in enumerative geometry, e.g., in the proof of Konstevich's formula for the number of rational plane curves passing through a fixed number of points. However, once the genus of the curve is greater than 0, the stable maps spaces become highly singular, having multiple irreducible components of varying dimensions. In order to have more refined enumerative counts, one tries to find desingularizations of these spaces. In this talk, I'll discuss the history of this desingularization problem for genus 1 stable maps to projective space, and conclude with a discussion of a new approach with the potential to work for higher genera (work in progress).
Stable maps from genus 1 curves to projective space
In this talk, we look at the classification problem for symmetric fusion categories in positive characteristic. We recall the second Adams operation on the Grothendieck ring and use its properties to obtain some classification results. In particular, we show that the Adams operation is not the identity for any non-trivial symmetric fusion category. We also give lower bounds for the rank of a (non-super-Tannakian) symmetric fusion category in terms of the characteristic of the field. As an application of these results, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects.