Factorial threefolds

阶乘三倍

• 批准号: EP/E048412/1
• 负责人: Ivan Cheltsov
• 金额: $ 23.75万
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2007
• 资助国家: 英国
• 起止时间: 2007 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE048412%2F1
• 关键词: Factorial; threefolds

Suppose that S is a finite set of points in n-dimensional space. In fields ranging from applied mathematics (splines and interpolation) to transcendental numbers, and of course also in algebraic geometry, it is interesting to ask when the points of the set S impose independent linear conditions on polynomials of degree at most d. This question has a long history. For example, in the case when n=2 and d=3, the answer to the question is given by the classical theorems of Pappus, Pascal and Chasles. There are of course many variants of the question. Perhaps, the most basic and useful is to take points in n-dimensional complex projective space and to ask about homogeneous forms of degree d instead of polynomials of degree at most d. The Cayley-Bacharach theorem may be seen as a partial answer to the above question.In commutative algebra, a unique factorization domain, or simply UFD, is a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology ofBourbaki. Most rings familiar from elementary mathematics are UFDs: the integers, the polynomial rings over a field, the formal power series ring over a field, the ring of functions in a fixed number of complex variables holomorphic at the origin etc.However, most factor rings of a polynomial ring are not UFDs. Fora given commutative ring, it is an interesting question to decide weather it is UFD or not.Two questions above are of different natures. However, they are closely related through the topology of mildly singular algebraic threefolds.In mathematics, an affine algebraic variety is essentially a set of common zeroes of a set of polynomials. Similarly, a projective algebraic variety is a set of common zeroes of a set of homogeneous forms. Algebraic varieties are one of the central objects of study in classical and modern algebraic geometry. An affine algebraic variety is called factorial if its coordinate ring is UFD. For a projective algebraic variety, one can define the factoriality in a similar way. In most of cases, the factoriality of projective varieties can be expressed in terms of topological data and can be proved by using powerful tools of topology such as the Lefschetz theorem and the Poincare duality.Algebraic surfaces, i.e., algebraic varieties of complex dimension two, are usually not factorial. For most of complex projective threefolds, i.e., algebraic varieties of complex dimension three, the factoriality simply means that its topology is trivial outside of the cycles of real dimension three. For example, every smooth threefold hypersurface is factorial by the Lefschetz theorem and the Poincare duality. For threefolds with isolated singularities, we still can use the Lefschetz theorem, but the Poincare duality usually fails. For example, every smooth threefold hypersurface is factorial if and only if the Poincare duality does not fail for it.For a wide class of singular threefolds, the factoriality problem was investigated by Clemens. He showed that the factoriality of many singular threefolds can beexpressed in terms of the number of independent linear conditions that their singular points impose on the homogeneous forms of certain degree. We plan to investigate how the factoriality of singular threefolds depends on the number of singular points. This problem can be studied by methods of commutative algebra, topology, differential geometry and algebraic geometry. We expect to obtain new and interesting results in this direction.Non-factorial threefolds are of great interest for algebraic geometry, topology, differential geometry and mathematical physics. We plan to investigate the geometry of non-factorial threefolds studied by Corti, Catanese, Reid andtheir students in order to obtain a counter-example to the conjecture of Corti on birational rigidity.

假设S是n维空间中的一个有限点集。从应用数学(样条和插值法)到超越数，当然还有在代数几何中，有趣的问题是，集合中的点何时对至多d次多项式施加独立的线性条件。这个问题由来已久。例如，在n=2和d=3的情况下，问题的答案由Pappus、Pascal和Chasles的经典定理给出。当然，这个问题有很多不同的说法。也许，最基本和最有用的是在n维复射影空间中取点，并询问d次齐次形式而不是至多d次多项式。Cayley-Bacharach定理可以被视为对上述问题的部分回答。在交换代数中，唯一因式分解域，或简称UFD，是一个交换环，其中每个元素都可以唯一地写成素元的乘积，类似于整数的算术基本定理。UFD有时称为阶乘环，遵循Bourbaki的术语。初等数学中常见的大多数环都是UFD：整数环、域上的多项式环、域上的形式幂级数环、在原点全纯的固定数目复变元函数环等。然而，多项式环的大多数因子环不是UFD。对于给定的交换环，判定它是否是UFD是一个有趣的问题。以上两个问题具有不同的性质。在数学上，仿射代数簇本质上是一组多项式的公共零点的集合。类似地，射影代数簇是一组齐次形式的公共零点的集合。代数簇是古典和现代代数几何的中心研究对象之一。一个仿射代数簇称为阶乘，如果它的坐标环是UFD。对于射影代数簇，可以用类似的方法定义阶乘性。在大多数情况下，射影簇的阶乘性可以用拓扑数据来表示，并且可以用Lefschetz定理和Poincare对偶等强大的拓扑学工具来证明，而代数曲面，即复维代数簇通常不是阶乘的。对于大多数复射影三重，即复数维3的代数变体，阶乘性仅仅意味着它的拓扑在实数维3的圈之外是平凡的。例如，每个光滑的三重超曲面都是Lefschetz定理和Poincare对偶的阶乘。对于具有孤立奇点的三重问题，我们仍然可以使用Lefschetz定理，但Poincare对偶通常是失败的。例如，每个光滑的三重超曲面是阶乘的当且仅当Poincare对偶不会失败。对于一大类奇异的三重超曲面，Clemens研究了阶乘问题。他证明了许多奇异三重数的阶乘性可以用它们的奇点施加在一定程度齐次形式上的独立线性条件的个数来表示。我们计划研究奇异三重数的阶乘性如何依赖于奇点的数目。这个问题可以用交换代数、拓扑学、微分几何和代数几何的方法来研究。我们期待着在这个方向上得到新的有趣的结果。非阶乘三重在代数几何、拓扑学、微分几何和数学物理中都有很大的兴趣。我们计划研究Corti，Catanes，Reid和他们的学生所研究的非阶乘三重的几何，以获得对Corti关于双胞胎刚性的猜想的反例。

项目成果

Exceptional del Pezzo hypersurfaces

卓越的 del Pezzo 超曲面

• DOI: 10.48550/arxiv.0810.2704
• 发表时间: 2008
• 作者: Cheltsov I

On exceptional quotient singularities

关于异常商奇点

• DOI: 10.48550/arxiv.0909.0918
• 发表时间: 2009
• 作者: Cheltsov I

Five embeddings of one simple group

一个简单组的五次嵌入

• DOI: 10.1090/s0002-9947-2013-05768-6
• 发表时间: 2013
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Cheltsov I

Del Pezzo Zoo

德尔佩佐动物园

• DOI: 10.1080/10586458.2013.813775
• 发表时间: 2013
• 期刊: Experimental Mathematics
• 影响因子: 0.5
• 作者: Cheltsov I

Six-dimensional exceptional quotient singularities

六维异常商奇点

• DOI: 10.4310/mrl.2011.v18.n6.a6
• 发表时间: 2011
• 期刊: Mathematical Research Letters
• 影响因子: 1
• 作者: Cheltsov I