Local & Global Problems

9400753 库特科斯基 主要研究者将研究代数几何中的四个问题。 他将致力于黎曼-罗克问题，这是计算上同调组的高倍数的线丛的投影品种。 他还将工作的地方模拟这一问题是计算希尔伯特函数的价值理想所确定的消失，以高多重沿着固定估值。 此外，主要研究者将研究分歧和等价的同态模。 本研究涉及交换代数和代数几何中的一些问题。 代数几何研究多项式方程族的解。 人们既可以研究解集的几何，也可以通过研究形成交换环的解集上的某些函数来代数地处理问题。 这种双重视角在交换代数和代数几何之间建立了密切的联系。

9400753 Cutkosky The principle investigator will investigate four problems in algebraic geometry. He will work on the Riemann-Roch problem which is to compute the cohomology groups of high multiples of a line bundle on a projective variety. He will also work on the local analogue of this problem which is the computation of Hilbert functions of valuation ideals determined by vanishing to high multiplicity along fixed valuations. In addition, the principle investigator will study ramification and the equivalence of homomorphisms of modules. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.