Abstract

The complex projective line CP1 is the only compact Riemann surface on which every degree 0 divisor is principal. For compact Riemann surfaces of higher genus, it is thus natural to ask how often does a ‘random’ divisor on the surface become principal.

Generalising classical work of Abel—Jacobi and being fairly liberal with what the word ‘solution’ means, the above problem admits a partial solution in the form of a distinguished homology class—- the double ramification (DR) cycle—- on the moduli space of Riemann surfaces. The latter space is not compact, and thus to obtain a more complete solution we need to extend the DR cycle to a compactification of the space.

In this talk, I will recall how the DR cycle is constructed, and how it can be extended using logarithmic geometry following the works of Marcus—Wise. I will also highlight some of its downstream enumerative applications. If time permits, I will attempt to shed some light on the appearance of the word ‘punctured’ in the title.