Let ${\mathbb{F}}_e$ denote the Hirzebruch surface $\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^1}\oplus {\mathcal{O}}_{{\mathbb{P}}^1}(e))$, and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves $M_{{\mathbb{F}}_e,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one.

We then study sharp Bogomolov inequalities $\mathrm{\Delta }\ge {\delta }_H(c_1/r)$ for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function ${\delta }_H(c_1/r)$ can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and ${\delta }_H(c_1/r)$ is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute ${\delta }_H(c_1/r)$ exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of $M_{{\mathbb{F}}_e,H}(\mathbf{v})$.

让 ${\mathbb{F}}_{Ë}$ 表示Hirzebruch表面 $\mathbb{P}（{\mathcal{Ø}}_{{\mathbb{P}}^{\mathrm{1个}}}\oplus {\mathcal{Ø}}_{{\mathbb{P}}^{\mathrm{1个}}}（Ë））$，并将H设为任意除数。在本文中，我们通过算法确定半稳定绳轮的模量空间${\mathrm{中号}}_{{\mathbb{F}}_{Ë}，H}（\mathrm{[R}，C_{\mathrm{1个}}，C_2）$是非空的。我们的算法依赖于某些优先滑轮。我们首先解决这些堆叠的存在问题，然后通过算法确定堆叠中一般捆的Harder-Narasimhan过滤。特别地，当且仅当Harder-Narasimhan过滤的长度为1时，才存在半稳定的滑轮。

然后，我们研究尖锐的Bogomolov不等式 $\mathrm{\Delta }\ge {\delta }_H（C_{\mathrm{1个}}/\mathrm{[R}）$对于考虑到极化和斜率的稳定滑轮的判别；这些不等式实质上完全描述了稳定槽轮的特性。功能${\delta }_H（C_{\mathrm{1个}}/\mathrm{[R}）$可以通过限制程序以任意精度计算。对于反规范极化的del Pezzo表面，特殊的束始终稳定且${\delta }_H（C_{\mathrm{1个}}/\mathrm{[R}）$由例外捆绑计算。更一般地说，我们表明，对于任意极化，除了稳定的特殊束所提供的那些之外，还存在稳定的滑轮的必要条件。我们计算${\delta }_H（C_{\mathrm{1个}}/\mathrm{[R}）$正是在某些情况下。最后，存在问题的解决方案可立即应用于图的两边几何。${\mathrm{中号}}_{{\mathbb{F}}_{Ë}，H}（\mathbf{v}）$。