论文标题
矢量束的模量空间上的理性曲线
Rational Curves on Moduli Spaces of Vector Bundles
论文作者
论文摘要
我们彻底描述了Moduli Space $ {\ rm su} _ {c}（c}（r，l）$ r $和curve $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ c $ r $的固定度$ k $ $ k $的希尔伯特方案的预期尺寸组成部分。我们表明，每$ k \ geq 1 $都有$ {\ rm gcd}（r，°L）$ unobstructed组件。此外，如果$ k $由$ r_1（r-r_1）（g-1）$以$ 1 \ le r_1 \ le r-1 $除外，则每个这样的$ r_1 $都有预期尺寸的额外阻塞组件。我们构建了阻塞组件的家庭，并表明它们的通用点不是给定等级和决定因素的通用矢量束。最后，我们还获得了$ {\ rm su} _ {c}（r，l）$的理性连接度的上限，该$ {\ rm su} _ {c}（r，l）$在维度中是线性的。
We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We show that for every $k \geq 1$ there are ${\rm gcd}(r, °L)$ unobstructed components. In addition, if $k$ is divisible by $r_1(r-r_1)(g-1)$ for $1\le r_1\le r-1$, there is an additional obstructed component of the expected dimension for each such $r_1$. We construct families of obstructed components and show that their generic point is not the generic vector bundle of given rank and determinant. Finally, we also obtain an upper bound on the degree of rational connectedness of ${\rm SU}_{C}(r,L)$ which is linear in the dimension.
