Positivity properties of toric line bundles and tropical divisors

代数簇是空间的一个子集，它是作为有限个多项式方程消失的点集给出的。这个定义的本质是各种几何性质和定义方程的代数性质之间的相互作用，这是代数几何中的一个经典和基本问题。本提案中描述的项目将为理解这一关系迈出重要的一步。希尔伯特意识到，研究各种定义方程的一种有效方法是考虑所谓的更高合度。第一个合并式是给定方程所满足的多项式关系。有有限多个第一合子生成(允许多项式系数)给定的一组方程的所有第一合子。现在，我们可以继续研究第一个合子和所谓的第二合子之间的关系，等等。Betti数记录了产生所有合子所需的一定程度的第i个合子的数目。它们携带了大量关于品种嵌入的信息。例如，假设方程在空间中的七个点上消失，人们可以从它们的Betti数中检测是否存在一个三次多项式在所有这些点上消失。通常，人们从抽象的角度研究代数簇，允许许多不同的嵌入到空间中，这些嵌入对应于所谓的非常充足的因子，即一维以下的子簇的某些形式和。研究这些因子的几何性质与所得到的嵌入的代数性质之间的关系是代数几何中的一个基本问题。几何性质的一个例子是除数与任意给定曲线相交的点数，而代数性质的一个例子是所有定义方程都可以由二次方程生成。这项建议涉及使用离散几何方法来研究这种关系。

An algebraic variety is a subset of space that is given as the set of points where finitely many polynomial equations vanish. Intrinsic in this definition is an interplay between geometric properties of the variety and algebraic properties of the defining equations, a classical and fundamental problem in algebraic geometry. The projects described in this proposal will provide a major step towards the understanding of this relationship. Hilbert realised that a powerful way of studying the defining equations of a variety is by considering the so-called higher syzygies. A first syzygy is a polynomial relations satisfied by the given equations. There are finitely many first syzygies that generate (allowing polynomial coefficients) all first syzygies of a given set of equations. Now one can continue by looking at the relations between the first syzygies, the so-called second syzygies, and so on. The Betti numbers record the number of ith syzygies of a certain degree needed to generate all syzygies. They carry a lot of information about the embedding of the variety. For example, given the equations vanishing on seven points in space, one can detect from their Betti numbers whether there exists a polynomial of degree 3 vanishing on all of them. Usually one studies algebraic varieties from the abstract point of view, admitting many different embeddings into space that correspond to so-called very ample divisors, certain formal sums of subvarieties of one less dimension. It is a fundamental question in algebraic geometry to study the relationship between geometric properties of these divisors and algebraic properties of the resulting embedding. An example of a geometric property would be in how many points the divisor intersects an arbitrary given curve, and an example of an algebraic property would be that all defining equations can be generated from equations of degree two. This proposal deals with investigating this relationship using methods of discrete geometry.