Relations in the tautological ring of the universal curve

Abstract

After some background theory we provide a brief summary of what is known about the tautological ring of the moduli space curves. We then formulate a few conjectures about the structure of the tautological ring of the universal curve. These conjectures are analogous to the so-called "Faber conjectures". We verify these conjectures for genus 2 < g < 9. We also study some matrices associated to the conjectures and find a realtionship between these matrices and the corresponding matrices on the moduli space of curves.