The moduli space of odd spin curves of genus 9 is unirational, realized birationally as a locally trivial projective bundle over a finite quotient of the moduli space of n-pointed odd stable spin curves of lower genus.
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The class of the bi-elliptic Prym locus RB_8^0 is shown to be non-tautological in CH^*(R_8), with a similar result for compact spaces when g + m >= 8.
Using the Chenevier-Lannes classification of automorphic representations and a conjectural correspondence to ℓ-adic Galois representations, the Euler characteristics of overline M_{3,n} and M_{3,n} (n≤14) and of local systems V_λ on A_3 (|λ|≤16) are computed in the Grothendieck group of such Galois/
Expository survey of when Chow rings and cohomology rings of moduli spaces of curves are tautological and when their point counts over finite fields are polynomials in q.
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The unirationality of $S_9^-$ and moduli spaces of pointed spin curves
The moduli space of odd spin curves of genus 9 is unirational, realized birationally as a locally trivial projective bundle over a finite quotient of the moduli space of n-pointed odd stable spin curves of lower genus.
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Non--tautological cycles on Prym moduli spaces
The class of the bi-elliptic Prym locus RB_8^0 is shown to be non-tautological in CH^*(R_8), with a similar result for compact spaces when g + m >= 8.
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Cohomology of moduli spaces via a result of Chenevier and Lannes
Using the Chenevier-Lannes classification of automorphic representations and a conjectural correspondence to ℓ-adic Galois representations, the Euler characteristics of overline M_{3,n} and M_{3,n} (n≤14) and of local systems V_λ on A_3 (|λ|≤16) are computed in the Grothendieck group of such Galois/
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Chow rings, cohomology rings, and point counts of moduli spaces of curves
Expository survey of when Chow rings and cohomology rings of moduli spaces of curves are tautological and when their point counts over finite fields are polynomials in q.