Threefold hypersurfaces with zero defect

수학강연회 5555 2025.04.14 16:18

https://www.youtube.com/embed/CGcTBPrOKpM + 750

강연자	홍규식
소속	전주대학교

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 of Bourbaki. 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. For a 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. The factoriality of threefolds with isolated singularities has been considered by several authors for a long time. For a wide class of singular threefolds, the factoriality problem, introducing the concept of defect, was investigated by Clemens. He showed that the factoriality of many singular threefolds can be expressed 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.