Constructions of nontautological classes on moduli spaces of curves

TL;DR

This paper constructs explicit algebraic cycles on moduli spaces of curves that are outside the tautological ring, providing new examples and a method for intersection computations.

Contribution

It introduces explicit non-tautological classes on moduli spaces and a general approach for intersection calculations within the tautological ring.

Findings

Explicit non-tautological classes constructed for large genus g and for M_2,20.

A general method for computing intersections in the tautological ring.

New examples demonstrating the complexity of algebraic cycles on moduli spaces.

Abstract

We construct explicit examples of algebraic cycles in \bar M_g (for large g congruent to 2 mod 4) and in M_2,20 (no bar) which are not in the tautological ring. In an appendix we give a general method for computing intersections in the tautological ring.