Green's Conjecture: a new approach via Chow forms of Grassmannians

I will discuss and prove an algebraic statement concerning the vanishing of the Koszul modules associated to any subspace inside the second exterior product of a complex vector space. This statement turns out to be equivalent to Mark Green's Conjecture on syzygies of canonical curves of genus g, which gives an alternative to Voisin's proof (in characteristic zero) and a first proof of Green's conjecture in characteristic p>(g+3)/2. I will also present topological applications of this result, focusing on an explicit description of the Cayley-Chow form of the Grassmannian of lines. This is joint work with M. Aprodu, S. Papadima, C. Raicu and J. Weyman.