Tahereh Aladpoush
Postulation of projective schemes supported on generic linear configurations: the case of lines

Studying the postulation of a projective scheme X means to determine the number of conditions imposed by asking hypersufaces to contain X. This is a well-known problem in Algebraic Geometry which reveals interesting aspects and unexpected information of the projective geometry of X. Indeed, there is an “expected” number of conditions- in this situation X is said to have good postulation. So a natural question to ask is: when does X have good postulation, and if it is not, why not? In the case when X is a scheme supported on generic configuration of linear spaces there is much interest in this question because of many applications inside and outside of mathematics. This problem can be traced back in classical projective geometry, but the first complete result was achieved in 1981 by R. Hartshorne and A. Hirschowitz on the postulation of generic lines. After that, despite all the progress made on this problem, very little is known about it and the problem is still widely open.

In the first part of this talk we introduce the postulation problem, then we discuss the problem for the family of schemes X supported on generic linear configurations in projective space. In the second part of this talk we focus on the case of reduced & non-reduced lines arrangements. Our main result provides a complete answer to the postulation problem for generic configurations of lines union one double line, which says that X always has good postulation except for just one case in 4-dimensional projective space. Moreover, we discuss an approach to the general case of lines union one multiple line by proposing a conjecture.