# An action of the Polishchuk differential operator via punctured surfaces

## TL;DR

This paper explores a new lifting of the Polishchuk differential operator acting on tautological algebras of Jacobians of pointed curves, linking it to punctured Riemann surfaces and tautological relations on moduli spaces.

## Contribution

It introduces a novel lifting of the Polishchuk operator to punctured surfaces and connects tautological relations to universal Jacobian relations.

## Key findings

- Proved tautological relations originate from universal Jacobian relations.
- Defined a lifting of the Polishchuk operator for punctured Riemann surfaces.
- Established a link between tautological algebra actions and moduli space relations.

## Abstract

For a family of Jacobians of smooth pointed curves there is a notion of tautological algebra. There is an action of $\mathfrak{sl}_2$ on this algebra. We define and study a lifting of the Polishchuk operator, corresponding to $f\in \mathfrak{sl}_2$, on an algebra consisting of punctured Riemann surfaces. As an application we prove that a collection of tautological relations on moduli of curves, discovered by Faber and Zagier, come from a class of relations on the universal Jacobian.