标题： 射影空间的二重覆盖上的Ulrich丛 显示英文标题

标题： Ulrich bundles on double coverings of projective space

摘要： 固定一个极化簇$X$，我们可以询问它是否具有Ulrich丛，如果有的话，它们的最小可能秩是多少。在本论文中，回顾了Ulrich层的一般性质后，我们证明了任何嵌入到权为$(1^{n+1},m)$的加权射影空间中的作为子簇的有限覆盖$\mathbb{P}^n$都具有Ulrich层，这是通过使用矩阵分解来实现的。在这些簇中，我们专注于具有$n\ge3$的双覆盖。 通过Hartshorne--Serre对应关系（我们在此过程中进行回顾），我们证明了一般的$X$当且仅当$n=3$和$m=2,3,4$时， admits一个秩为$2$的Ulrich层，并表征了它们截面的零点。此外，我们构造了它们模空间的预期维数的通用光滑分支，分析了自然对合在这些分支上的作用以及这些丛在低次数超曲面上的限制。对于$m=2,3$，我们验证了所有可能秩的斜稳定Ulrich丛的存在性。 显示英文摘要

摘要： Fixed a polarised variety $X$, we can ask if it admits Ulrich bundles and, in case, what is their minimal possible rank. In this thesis, after recalling general properties of Ulrich sheaves, we show that any finite covering of $\mathbb{P}^n$ that embeds as a divisor in a weighted projective space with weights $(1^{n+1},m)$ admits Ulrich sheaves, by using matrix factorisations. Among these varieties, we focus on double coverings of with $n\ge3$. Through Hartshorne--Serre correspondence, which we review along the way, we prove that the general such $X$ admits a rank $2$ Ulrich sheaf if and only if $n=3$ and $m=2,3,4$, and characterise the zero loci of their sections. Moreover, we construct generically smooth components of the expected dimension of their moduli spaces, analyse the action of the natural involution on them and the restriction of those bundles to low degree hypersurfaces. For $m=2,3$, we verify the existence of slope-stable Ulrich bundles of all the possible ranks.