Factorial Threefolds

The main goal of this project is to study factoriality of nodal threefolds. Namely, this project is about finding new relations between the topology of a singular algebraic threefold (an algebraic variety of dimension three) and the number, types, and the mutual position of its singular points. Such threefold is called factorial if the Poincare duality holds for it, which basically means that its topology is very similar to the topology of smooth threefolds.

The Cayley–Bacharach theorem, in its classical form, may be seen as a result
about the number of independent linear conditions imposed on polynomials of a given degree by a certain finite set of points in the plane. This result has a long and interesting history, starting with a famous theorem of Pappus of Alexandria, and continuing with results of Pascal, Chasles, Cayler, Bacharach and Macaulay. As the methods of algebraic geometry have changed over the years, the result has been successfully generalized, improved, and interpreted. This development continues today, and the funded project is a part of this development.

Several generalizations of the classical theorem of Cayley and Bacharach were proved. One application gave a proof of the long standing conjecture of Ciliberto about the factoriality of nodal threedimensional hypersurfaces (it basically states that three-dimensional hypersurface of degree with simplest ordinary double points is factorial if it has at most (d-1)(d-1)-1 singular points and this bound is sharp).