论文标题
Ulrich束的Koszul属性和稳定束的模量空间的合理性在Del Pezzo表面
Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces
论文作者
论文摘要
令$ \ mathcal {e} $为平滑的投影品种上的矢量捆绑包$ x \ subseteq \ mathbb {p}^n $，相对于超平面部分$ h $，这是Ulrich。在本文中，我们研究了$ \ Mathcal {e} $的Koszul属性，$ k $ th的slope-themtemthesemessissiblesyssemessemistable syzygy bundle $ \ mathcal {s} _k（\ nathcal {e} $均可用于所有$ k \ geq 0 $ and gemational and slope of slope bundles bund of slope bund bund。由于我们的研究结果，我们表明，如果$ x $是$ d \ geq 4 $的del pezzo表面，那么任何Ulrich Bundle $ \ MATHCAL {E} $都可以满足Koszul属性，并且可以坡度稳定。我们还表明，对于无限的许多Chern字符$ {\ bf v} =（r，c_1，c_2）$，相应的Slope-stable捆绑包$ \ mathfrak {m} _h（{\ bf v}）$的模量空间是有理由的，因此是有理由的，并因此而产生新的evurer。结果，我们表明，在这些模量空间中，Ulrich束的迭代式束束很密集。
Let $\mathcal{E}$ be a vector bundle on a smooth projective variety $X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section $H$. In this article, we study the Koszul property of $\mathcal{E}$, the slope-semistability of the $k$-th iterated syzygy bundle $\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.
