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Affine pavings for moduli spaces of pure sheaves on \(\mathbb {P}^2\) with degree \(\le 5\)

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Let \(M(d,r)\) be the moduli space of semistable sheaves of rank 0, Euler characteristic \(r\) and first Chern class \(dH~(d>0)\), with \(H\) the hyperplane class in \(\mathbb {P}^2\). In [14] we gave an explicit description of the class \([M(d,r)]\) of \(M(d,r)\) in the Grothendieck ring of varieties for \(d\le 5\) and \(g.c.d(d,r)=1\). In this paper we compute the fixed locus of \(M(d,r)\) under some \((\mathbb {C}^{*})^2\)-action and show that \(M(d,r)\) admits an affine paving for \(d\le 5\) and \(g.c.d(d,r)=1\). We also pose a conjecture that for any \(d\) and \(r\) coprime to \(d\), \(M(d,r)\) would admit an affine paving.

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Acknowledgments

I was partially supported by NSFC grant 11301292. When I wrote this paper, I was a post-doc at MSC in Tsinghua University in Beijing. Finally I thank Y. Hu for some helpful discussions.

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  1. MSC, Tsinghua University, Beijing, 100084, China

    Yao Yuan

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  1. Yao Yuan
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Yuan, Y. Affine pavings for moduli spaces of pure sheaves on \(\mathbb {P}^2\) with degree \(\le 5\) . Geom Dedicata 177, 385–400 (2015). https://doi.org/10.1007/s10711-014-9995-x

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