Some Betti numbers of the moduli of 1-dimensional sheaves on \(\mathbb{P}^2\). (English) Zbl 1541.14017
Summary: Let \(M(d,\chi)\), with \((d,\chi)=1\), be the moduli space of semistable sheaves on \(\mathbb{P}^2\) supported on curves of degree \(d\) and with Euler characteristic \(\chi\). The cohomology ring \(H^*(M(d,\chi),\mathbb{Z})\) of \(M(d,\chi)\) is isomorphic to its Chow ring \(A^*(M(d,\chi))\) by
E. Markman’s result [Adv. Math. 208, No. 2, 622–646 (2007; Zbl 1115.14036)].
W. Pi and J. Shen [Algebr. Geom. 10, No. 4, 504–520 (2023; Zbl 1528.14016)]
have described a minimal generating set of \(A^*(M(d,\chi))\) consisting of \(3d-7\) generators, which they also showed to have no relation in \(A^{\leq d-2}(M(d,\chi))\). We compute the two Betti numbers \(b_{2(d-1)}\) and \(b_{2d}\) of \(M(d,\chi)\), and as a corollary we show that the generators given by Pi and Shen have no relations in \(A^{\leq d-1}(M(d,\chi))\), but do have three linearly independent relations in \(A^d(M(d,\chi))\).
Keywords:
moduli spaces of semistable 1-dimensional sheaves on projective surfaces; motivic measures; Betti numbers; generators of the Chow ringsReferences:
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