Geodesic
Cohomological $\chi $-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$
Forum of Mathematics, Sigma, Tome 12 (2024)

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We prove that the cohomology rings of the moduli space $M_{d,\chi }$ of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the $\chi $-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that $M_{d,\chi }$ are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties. 
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     title = {Cohomological $\chi $-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$},
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Woonam Lim; Miguel Moreira; Weite Pi. Cohomological $\chi $-dependence of ring structure for the moduli of one-dimensional sheaves on $\mathbb {P}^2$. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.31

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