论文标题
关于P^2
Higher rank Brill-Noether theory on P^2
论文作者
论文摘要
令$ m _ {\ mathbb {p}^2}（v）$为$ \ mathbb {p}^2 $上的可半轴的模块空间，然后让$ b^k（v）\ subseteq m _ { $ h^0（\ mathbb {p}^2，e）\ geq k $。在本文中，我们在$ \ mathbb {p}^2 $上开发了Brill-Noether Loci的基础属性。设置$ r = r（e）$作为等级，$ C_1，C_2 $ CHERN类。 Brill -Noether基因座具有自然的确定性方案结构和预期尺寸$ dim b^k（v）= dim m _ {\ mathbb {p}^2}^2}（v）（v） - k（k-χ（e））$。当$ C_1> 0 $时，我们表明Brill-Noether locus $ b^r（v）$是非空的。当$ C_1 = 1 $时，我们显示所有Brill-Noether Loci都是不可约的，并且是预期的维度。我们表明，当$μ= C_1/R> 1/2 $不是整数，而$ C_2 \ gg 0 $时，Brill-Noether Loci可以还原，并描述预期和意外尺寸的明显不可减至的组件。
Let $M_{\mathbb{P}^2}(v)$ be a moduli space of semistable sheaves on $\mathbb{P}^2$, and let $B^k(v) \subseteq M_{\mathbb{P}^2}(v)$ be the \textit{Brill-Noether locus} of sheaves $E$ with $h^0(\mathbb{P}^2, E) \geq k$. In this paper we develop the foundational properties of Brill-Noether loci on $\mathbb{P}^2$. Set $r = r(E)$ to be the rank and $c_1, c_2$ the Chern classes. The Brill-Noether loci have natural determinantal scheme structures and expected dimensions $dim B^k(v) = dim M_{\mathbb{P}^2}(v) - k(k - χ(E))$. When $c_1 > 0$, we show that the Brill-Noether locus $B^r(v)$ is nonempty. When $c_1 = 1$, we show all of the Brill-Noether loci are irreducible and of the expected dimension. We show that when $μ= c_1/r > 1/2$ is not an integer and $c_2 \gg 0$, the Brill-Noether loci are reducible and describe distinct irreducible components of both expected and unexpected dimension.
