Logarithmic approach to the double ramification cycle

Schwarz, R.M. (2023)

Doctoral Thesis

This thesis discusses several questions regarding the double ramification cycle as a Chow class on the moduli space of stable n-pointed genus g curves using tools from so-called logarithmic geometry. It contains two extracts from articles; the first of these defines the universal double ramification cycle on the Picard stack of n-pointed genus g curves with a line bundle of fixed degree, which is a way to also include the generalisations that are called twisted double ramification cycles. The second article introduces the logarithmic double ramification cycle in the logarithmic Chow ring. The logarithmic double ramification cycle is proven to be ‘logarithmically tautological’ and it helps us prove that the double-double ramification cycle (or the good definition for ‘intersecting double ramification cycles’) is tautological – that is, these classes lie in a subring generated by ‘computable and known’ classes. The second chapter of the thesis explains and illustrates piecewise-polynomial functions, which are key to describing the forementioned ‘logarithmically tautological’, and how these functions relate to classical divisors which we use to describe tautological rings.

• Algebraic geometry
• Moduli space of stable curves
• Logarithmic geometry
• Double ramification cycle Tautological rings
• Picard stack
• Logarithmic Chow ring
• Piecewise-polynomial functions

• All authors: Schwarz, R.M.
• Supervisor: Holmes, D.S.T.; Luijk, R.M. van; Edixhoven, S.J.†
• Committee: Derks, G.L.A.; Hupkes, H.J.; Cavalieri, R.; Carocci, F.; Sauvaget, A.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Faculty of Science, Leiden University
• Date: 2023-12-07