摘 要：代数几何作为现代数学的重要分支，在曲线与曲面分类研究中发挥着不可替代的作用。本文以代数几何理论为基础，聚焦于低维代数簇的拓扑结构与几何性质，旨在通过引入新型不变量体系，建立更加精细的分类框架。研究采用Hodge理论、层论及模空间理论等先进工具，对复射影空间中的代数曲线和代数曲面进行系统分析。通过对经典Chow环理论的拓展应用，首次提出基于特征类的全新分类方法，解决了传统方法在高亏格情形下的局限性问题。实验结果表明，该方法能够有效区分具有相同拓扑类型的代数簇，并揭示其内在几何差异。本研究不仅为代数几何领域提供了新的研究视角，还为计算机辅助几何设计等相关学科提供了坚实的理论基础，特别是在自由曲面造型、计算机视觉等领域展现出广阔的应用前景。研究成果丰富了代数几何理论体系，为后续研究奠定了重要基础。
关键词：代数几何；低维代数簇；特征类分类
Abstract：Algebraic geometry, as a crucial branch of modern mathematics, plays an indispensable role in the classification of curves and surfaces. This paper, grounded in algebraic geometry theory, focuses on the topological structures and geometric properties of low-dimensional algebraic varieties, aiming to establish a more refined classification fr amework by introducing a novel system of invariants. Advanced tools such as Hodge theory, sheaf theory, and moduli space theory are employed to systematically analyze algebraic curves and surfaces in complex projective spaces. By extending the classical Chow ring theory, this study proposes for the first time a new classification method based on characteristic classes, addressing the limitations of traditional methods in high-genus cases. Experimental results demonstrate that this approach can effectively distinguish algebraic varieties with identical topological types while revealing their intrinsic geometric differences. This research not only provides a new perspective for the field of algebraic geometry but also offers a solid theoretical foundation for related disciplines such as computer-aided geometric design, particularly in areas like free-form surface modeling and computer vision, showcasing broad application prospects. The findings enrich the theoretical fr amework of algebraic geometry and lay an important foundation for future research.
引言 1
一、曲线分类的代数几何基础 1
(一)曲线的基本不变量 1
(二)代数曲线的奇点分析 2
(三)曲线的模空间理论 2
二、曲面分类中的拓扑不变量 3
(一)曲面的陈类与特征 3
(二)代数曲面的交截理论 3
(三)曲面的典范除子研究 4
三、代数几何在低维流形的应用 4
(一)低维流形的结构定理 4
(二)流形上的霍奇理论应用 5
(三)低维流形的分类方法 5
四、现代代数几何技术进展 6
(一)极小模型纲领介绍 6
(二)密克图与稳定曲线理论 6
(三)镜像对称的应用实例 7
结论 7
参考文献 9
致谢 9
关键词：代数几何；低维代数簇；特征类分类
Abstract：Algebraic geometry, as a crucial branch of modern mathematics, plays an indispensable role in the classification of curves and surfaces. This paper, grounded in algebraic geometry theory, focuses on the topological structures and geometric properties of low-dimensional algebraic varieties, aiming to establish a more refined classification fr amework by introducing a novel system of invariants. Advanced tools such as Hodge theory, sheaf theory, and moduli space theory are employed to systematically analyze algebraic curves and surfaces in complex projective spaces. By extending the classical Chow ring theory, this study proposes for the first time a new classification method based on characteristic classes, addressing the limitations of traditional methods in high-genus cases. Experimental results demonstrate that this approach can effectively distinguish algebraic varieties with identical topological types while revealing their intrinsic geometric differences. This research not only provides a new perspective for the field of algebraic geometry but also offers a solid theoretical foundation for related disciplines such as computer-aided geometric design, particularly in areas like free-form surface modeling and computer vision, showcasing broad application prospects. The findings enrich the theoretical fr amework of algebraic geometry and lay an important foundation for future research.
Keywords: Algebraic Geometry；Low-Dimensional Algebraic Varieties；Characteristic Class Classification
引言 1
一、曲线分类的代数几何基础 1
(一)曲线的基本不变量 1
(二)代数曲线的奇点分析 2
(三)曲线的模空间理论 2
二、曲面分类中的拓扑不变量 3
(一)曲面的陈类与特征 3
(二)代数曲面的交截理论 3
(三)曲面的典范除子研究 4
三、代数几何在低维流形的应用 4
(一)低维流形的结构定理 4
(二)流形上的霍奇理论应用 5
(三)低维流形的分类方法 5
四、现代代数几何技术进展 6
(一)极小模型纲领介绍 6
(二)密克图与稳定曲线理论 6
(三)镜像对称的应用实例 7
结论 7
参考文献 9
致谢 9
