Geodesic
Wall-crossing integral chow rings of ${\overline {\mathcal {M}}}_{1,n\leq 4}$
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e196

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We compute the integral Chow rings of $\overline {\mathcal {M}}_{1,n}$ for $n=3,4$. The alternative compactifications introduced by Smyth – and studied further by Lekili and Polishchuk – present each of these stacks as a sequence of weighted blow-ups and blow-downs from a weighted projective space. We compute all the integral Chow rings by repeated application of the blow-up formula. 
Battistella, Luca; Lorenzo, Andrea Di. Wall-crossing integral chow rings of ${\overline {\mathcal {M}}}_{1,n\leq 4}$. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e196. doi: 10.1017/fms.2025.10143
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