The classification of universal Jacobians over the moduli space of curves

TL;DR

This paper provides a complete birational classification of the universal Jacobian over the moduli space of curves, revealing how its geometric properties change with genus, especially at critical transition points g=10 and 11.

Contribution

It offers a detailed classification of the universal Jacobian's birational type across all genera, including new results on its Kodaira dimension and disproving previous expectations about its relation to the moduli space.

Findings

Universal Jacobian is unirational for g<10

Kodaira dimension is zero at g=10

Kodaira dimension is 19 at g=11

Abstract

We carry out a complete birational classification of the degree g universal Jacobian P_g over the moduli space of curves, highlighting the transition cases g=10, 11. The universal Jacobian is unirational when g<10, has Kodaira dimension zero for g=10 and Kodaira dimension 19 for g=11. For g>11, the variety P_g has Kodaira dimension 3g-3, that is, the maximum allowed by Iitaka's easy addition formula for fibre spaces. In particular, we disprove the expectation that P_g and M_g have the same Kodaira dimension for all genera.