Hilbert schemes of points and applications

Abstract

Long, long ago 1 , in a galaxy 2 far, far away 3 ...The following is the author's PhD Thesis, defended in September 2017. It has not been edited since (except for the footnotes in the open problems section and for this note). In particular, there is nothing here about applications of Białynicki-Birula decomposition to Hilbert schemes [Jel19, Jel20, SS21], the multigraded Hilbert schemes and border apolarity [BB21, Mań20], fiber-full Hilbert schemes [CRR21, CRR22] and other recent developments. Philosophically speaking, is it reassuring that there are too many new interesting papers on Hilbert schemes of points to summarize here. It is also very interesting that many of the open problems listed below in §1.5 remain actual. Today, the author would add also the more applied open problems (see [JLP22]): a large part of the classical problem of determining the complexity of matrix multiplication can be reduced fruitfully to the study of the Hilbert scheme of points.While the author is not aware of any errors, it is likely they exist here. This work is put on arXiv due/thanks to several people who decided to refer to it and only to grant this piece a quasi-permanent place to rest.1 That is, five years 2 Metaphorically speaking 3 In terms of scientific development ¨