Koszul Modules

The goal of these notes is to introduce the basic theory of Koszul modules, and survey some applications to syzygies and to Green’s conjecture, following recent work of Aprodu, Farkas, Papadima, Raicu, and Weyman (Invent. Math. 218(3), 657–720 (2019)). More precisely, we discuss a relationship between Koszul modules and the syzygies of the tangent developable surface to a rational normal curve, which arises from a version of Hermite reciprocity for SL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\operatorname {SL}_2$$ \end{document}-representations. We describe in terms of the characteristic of the underlying field which of the Betti numbers of this surface are zero and which ones are not. We indicate how, by passing to a hyperplane section and using degenerations, the generic version of Green’s conjecture can be deduced in almost all characteristics. These notes are accompanied by examples and exercises, and include an Appendix on multilinear algebra.