Oberwolfach May 2006

These are the notes I typed during the talks. I haven’t spent much time after the lectures cleaning them up so there are probably lots of typos and even some more serious errors. The hardest thing was getting the quiver diagrams done in real time. I often had to give up! Anyway, there may be some value to this so I’m placing it in the public domain. feedback is welcome. 1. King—Moduli of sheaves from moduli of Kronecker modules with Alvarez-Consul 1.1. Basics. Let X be a projective scheme over k = k̄. Important features of cohX: monoidal category with identity O, simple objects Ox, x ∈ X, and automorphisms (n) : E 7→ E(n) given by tensoring n times with an ample invertible sheaf O(1). Notation: • H = H(n0, n1) := H(O(n1 − n0)) = Hom(O(−n1),O(−n0)); typically n1 ≥ n0. • MX(P ):=moduli space of semistable sheaves with Hilbert polynomial P . It is a projective scheme. Motivating question: why is MX(P ) projective? Classical answer: because it is constructed by GIT. Not very helpful answer! 1.2. Modern answer. First, why is X projective? Because when n1 − n0 0 (i.e., ∃ N ∀ n0, n1 with n1 − n0 ≥ N ...) the map x 7→ φx : H(Ox(n0))⊗H → H(Ox(n1)) is an embedding X → P(H∗) = MH(1, 1)=the moduli space of strictly non-zero H-Kronecker modules! Theorem 1.1. MX(P ) is projective because for n1 n0 0 (i.e., ∃ N0 ∀ n0 ≥ N0 ∃ N1 ∀, n1 ≥ N1....) the functor E 7→ φE : H(E(n0))⊗H → H(E(n1)) gives an embedding MX(P ) → MH(P (n0), P (n1))=the moduli space of semistable H-Kronecker modules which is obviously (!) projective. Remarks: (1) The proof gives a “new” construction of the MX(P ) (cf. Simpson). (2) the embedding is scheme-theoretic, except possibly at strictly semistable points when char k = p; 1Representations of dimension (n0, n1) = (1, 1) of the Kronecker quiver with dimH arrows.