Title: Double Ramification cycles in the punctured setting
Speaker: Xuanchun Lu
Abstract: The projective line PP^1 is the only algebraic curve on which every degree 0 divisor is principal. For any other algebraic curve, it is thus natural to ask when a ‘random’ divisor on the said curve becomes principal.Generalising classical works of Abel–Jacobi, a solution to the above problem may be given as a collection of cycles on the moduli spaces $M_{g, n}$ of genus g, n-marked algebraic curves. These cycles are known as the double ramification (DR) cycles. Since $M_{g, n}$ is not compact, these cycles do not fully capture the geometry of our situation. We thus need to seek an extension of DR cycles to some compactification of $M_{g, n}$.
In this talk, I will explain some of the challenges involved in constructing such an extension, and how they can be resolved using ideas and techniques from logarithmic geometry. I will also highlight some downstream enumerative applications of DR cycles. Time permitting, I will attempt to shed some light on the appearance of the word ‘punctured’ in the title.
Some snacks will be provided before and after the talk.