Four Pivotal Problems in Classical Algebraic Geometry

9900025这个项目涉及代数几何中一些相当具体的问题，这些问题对该学科某些思想的发展至关重要。 这些是：五个变量的三次多项式的零点参数化的可能性;四个变量的三次多项式的零轨迹中包含的曲线族;三个变量的六次多项式的值是平方的轨迹的密度;以及在四元五次多项式的零轨迹上五次三重有理曲线族的存在性。代数几何是研究多项式方程组解的几何学。 通过分析多项式的代数，我们理解了它们的解集的几何;从这些集合的知识中，我们最终达到了对代数的更深入和更有效的理解。

9900025This project is concerned with a number of fairly specific questions in algebraic geometry that are pivotal to the development of certain ideas in the subject. These are: the possibility of parametrizing the zeroes of a cubic polynomial in five variables; the families of curves contained in the zero locus of a cubic polynomial in four variables; the density of the locus where the value of a sextic polynomial in three variables is a square; and the existence of families of rational curves on quintic threefolds in the zero locus of a quintic polynomial in four variables.Algebraic geometry is the study of the geometry of the sets of solutions to polynomial equations. By analyzing the algebra of polynomials, we come to understand the geometry of their solution sets; and from a knowledge of these sets we arrive ultimately at a deeper and more effective understanding of algebra.