Mathematical Sciences: Algebraic Geometry and Seiberg-Witten Invariants

弗里德曼 弗里德曼研究新的4流形不变量介绍了塞伯格和维滕和他们的连接与代数几何的情况下，代数曲面。 对于许多有趣的曲面，曲面的代数几何与Seiberg-Witten理论之间几乎没有联系。 然而，对于直纹曲面，Seiberg-Witten理论与经典代数几何中的某些问题有着密切的联系。 如果直纹曲面是一个乘积，那么塞伯格-威滕模空间的描述与曲线的布里-诺特理论有关，无论是在变形理论还是计数理论上。 如果直纹曲面更一般，则Seiberg-Witten模空间与代数曲线上的稳定秩二向量丛问题有关。 一个问题是一般直纹曲面的Hilbert格式是否光滑。 这个问题与芒福德的一个著名例子有关，芒福德说，一般直纹曲面上的每一条不可约曲线都有正自相交。 经典代数几何中的第二个问题是研究稳定丛的模空间中的各种枚举轨迹，其中希尔伯特方案不光滑或不具有预期的维数。 另一组问题涉及研究辛4-流形的分类，这在许多方面应该类似于代数曲面的分类。 代数几何是数学的一个古老的分支，它研究平面或空间中多项式方程的求解问题。 研究这类方程的复数解是很重要的，这样可以看到更多的解的几何形状。在过去的50年里，代数、分析和拓扑学的新方法在这一理论中产生了深刻的结果。 由于一个代数曲面依赖于两个复数，因此依赖于四个真实的数，它是一个四维对象，因此在物理上是重要的，因为时空的宇宙是四维的。 最近的物理学思想已经导致了代数曲面的拓扑结构与几何结构之间的深刻突破。 这些方法也导致了对四维物体的理解的进步，这些物体的几何形状与经典力学有关，被称为辛4-流形。

Friedman Friedman studies the new 4-manifold invariants introduced by Seiberg and Witten and their connections with algebraic geometry in the case of algebraic surfaces. For many interesting surfaces, there is little contact between the algebraic geometry of the surface and Seiberg-Witten theory. However, for ruled surfaces, there is a close connection between Seiberg-Witten theory and certain questions in classical algebraic geometry. If the ruled surface is a product, then the description of the Seiberg-Witten moduli spaces is related to Brill-Noether theory for curves, in both its deformation-theoretic and enumerative guises. If the ruled surface is more general, the Seiberg-Witten moduli spaces are connected to questions concerning stable rank two vector bundles over algebraic curves. One problem is whether the Hilbert scheme of a general ruled surface is smooth. This problem is related to a famous example of Mumford, which says that every irreducible curve on a general ruled surface has positive self-intersection. A second question in classical algebraic geometry is to study various enumerative loci in the moduli space of stable bundles where the Hilbert scheme is not smooth or does not have the expected dimension. Another set of questions involves studying the classification of symplectic 4-manifolds, which in many ways should be similar to the classification of algebraic surfaces. Algebraic geometry is an old branch of mathematics which studies the solutions to polynomial equations in the plane or in space. It is important to study the solutions in complex numbers to such equations, in order to see more of the geometry of the solutions. Over the last fifty years, new methods in algebra, analysis, and topology have led to deep results in this theory. Since an algebraic surface depends on two complex numbers and hence four real numbers, it is a four-dimensional object and thus is important physically, since the universe of space-time has four dimensions. Recent ideas from physics have led to deep breakthroughs relating the topology of an algebraic surface to its geometry. These methods have also led to progress in the understanding of four-dimensional objects with a geometry connected to classical mechanics, which are called symplectic 4-manifolds.