Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ²

2022/07/17 by Yuan, Yao
#14D22 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.08060

Abstract

Let M(d,χ) with (d,χ)=1 be the moduli space of semistable sheaves on ℙ² supported on curves of degree d and with Euler characteristic χ. The cohomology ring H^*(M(d,χ),ℤ) of M(d,χ) is isomorphic to its Chow ring A^*(M(d,χ)) by Markman's result. W. Pi and J. Shen have described a minimal generating set of A^*(M(d,χ)) consisting of 3d-7 generators, which they also showed to have no relation in A^{≥ d-2}(M(d,χ)). We compute the two Betti numbers b_2(d-1) and b_2d of M(d,χ) and as a corollary we show that the generators given by Pi-Shen have no relations in A^{≥ d-1}(M(d,χ)) but do have three linearly independent relations in A^d(M(d,χ)).

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