Hilbert–Burch Linear Sections

This chapter takes an exception to the previous chapters in that the focus is on linear sections of \(m\times (m-1)\) generic matrices instead of square ones. Because of the celebrated theorem of Hilbert–Burch associated to codimension 2 perfect ideals and such matrices, we call the latter by abuse Hilbert–Burch matrices. The first part deals with such matrices whose entries are general linear forms in a d-dimensional polynomial ring over a field with focus on the watershed case where \(m\geq d\) . In this situation the symbolic Rees algebra of the ideal of \((m-1)\) -minors is thoroughly discussed, and the full nature of the cases \(m=d\) and \(m=d+1\) is displayed. A notable subsumed Cremona map based on certain inversion factors is given full description. In the sequel \(m\times (m-1)\) Hankel linear sections are discussed with an emphasis on the related algebraic invariants. A large section is dedicated to Hilbert–Burch linear sections when \(d=3\) , where the central theme is the discussion of the chaos invariant, a number based on the behavior of the ideals of lower minors. Other notions are shown to be related, such as fat points and reciprocal ideals of hyperplane arrangements. The chapter ends with a thorough discussion of linearly presented monomial ideals in three variables.