发表机构
Temple University; University of Houston（天普大学; 休斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究代数曲面上稳定层扩张的稳定性与极大子层，通过扩张构造证明有理曲面上弱Brill--Noether定理，并细化Segre分层，在射影平面上证明线丛对应层不可约且维数正确，表明细化Segre层优于Brill--Noether层。
AI 中文摘要
我们研究代数曲面上的两个密切相关的话题：由稳定层通过稳定层扩张得到的层的稳定性，以及给定稳定层的极大子层。对于前者，我们通过扩张提供了层的完全族的构造，并在某些情形下证明了稳定性。该构造使我们能够在有理曲面上证明某些情形的弱Brill--Noether定理。第二个话题引出了层的模空间的Segre分层的一个细化。我们在有理曲面和K3曲面上获得了某些细化Segre层的预期维数。在射影平面上，我们证明了我们的主要结果：所有对应于线丛的细化Segre层都是不可约的，具有预期维数，并且在取闭包下是嵌套的。作为推论，我们获得了更多扩张稳定性的情形。我们的结果表明，细化Segre层的行为远优于Brill--Noether层。我们对细化Segre层的研究关键依赖于Bridgeland稳定性条件和Li与Zhao的工作。
英文摘要
We study two closely related topics over algebraic surfaces: stability of extensions of stable sheaves by stable sheaves, and maximal subsheaves of a given stable sheaf. For the former, we provide a construction of complete families of sheaves via extensions and prove the stability for some cases. The construction enables us to prove certain cases of Weak Brill--Noether over rational surfaces. The second topic leads to a refinement of the Segre stratification of the moduli of sheaves. We obtain the expected dimension of certain refined Segre strata over rational surfaces and K3 surfaces. Over the projective plane, we prove our main result that all refined Segre strata corresponding to line bundles are irreducible of the expected dimension and are nested under taking closures. As a consequence, we obtain more cases of the stability of extensions. Our results suggest that refined Segre strata behave much better than Brill--Noether strata. Our study of the refined Segre strata relies crucially on Bridgeland stability conditions and the work of Li and Zhao.
Comments36 pages
