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.