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

Abstract

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