This is probably well known to algebraic geometers... Brill-Noether theory gives us Brill-Noether divisors in $\overline{M}_g$. Are there any interesting cycles in $A^*(\overline{M}_{g,n})$ coming from Hurwitz theory?
Cycles in the Chow ring of the moduli of curves coming from Hurwitz theory
Nati
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