论文标题
$ \ mathbb {p}^2 $上的一些betti数字数量
Some Betti numbers of the moduli of 1-dimensional sheaves on $\mathbb{P}^2$
论文作者
论文摘要
令$ m（d，χ）$带有$（d，χ）= 1 $是$ \ mathbb {p}^2 $在度量$ d $的曲线和Euler特征$χ$上支撑的可半轴束的模量空间。 Markman的结果是$ m（d，χ）$的$ M（d，χ）$的同型$ h^*（m（d，χ），\ Mathbb {z}）$是同构的。 W. Pi和J. Shen描述了由$ 3D-7 $发电机组成的$ a^*（m（d，χ））$的最小生成集，它们也表明它们与$ a^{\ geq d-2}（m（m（d，χ））中没有关系。我们计算了两个贝蒂数字$ b_ {2（d-1）$和$ b_ {2d} $的$ m（d，χ）$，作为推论，我们表明，pi-shen给出的发电机在$ a^{\ geq d-1}（\ geq d-1}（m（d，χ）$中，但有三个lineal lineeal specialss $ n $ a^d d（m）
Let $M(d,χ)$ with $(d,χ)=1$ be the moduli space of semistable sheaves on $\mathbb{P}^2$ supported on curves of degree $d$ and with Euler characteristic $χ$. The cohomology ring $H^*(M(d,χ),\mathbb{Z})$ 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 $A^{\geq d-2}(M(d,χ))$. 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 $A^{\geq d-1}(M(d,χ))$ but do have three linearly independent relations in $A^d(M(d,χ))$.
