EXCEPTIONAL LOCI ON M 0,n AND HYPERGRAPH CURVES ANA-MARIA CASTRAVET AND JENIA TEVELEV Abstract. We give a myriad of examples of extremal divisors, rigid curves, and birational morphisms with unexpected properties for the Grothendieck–Knudsen moduli space M0,n of stable rational curves. The basic tool is an isomorphism between M0,n and the Brill–Noether locus of a very special reducible curve corresponding to a hypergraph. Contents §1. Introduction 1 §2. Hypergraph Curves and their Brill–Noether Loci 3 §3. Extremal Divisors on M 0,n 6 §4. Admissible Sheaves on Hypergraph Curves 12 §5. Product of Linear Projections 19 §6. Proof of Theorems 3.5 and 3.8 21 §7. Exceptional Curves on M 0,n 28 §8. Appendix (after Sean Keel & James McKernan [KM]) 36 References 38 §1. Introduction For any projective variety X , the basic gadgets encoding combinatorics of its birational geometry are the Mori cone NE 1 ⊂ NS ∗ R (the closure of the cone generated by effective 1-cycles) and the effective cone Eff ⊂ NS R (the closure of the cone generated by effective Cartier divisors). Here NS is the Neron-Severi group of X . Let X = M 0,n be the moduli space of stable rational curves with n marked points. It is stratified by the topological type of a stable rational curve and so it has “natural” boundary effective divisors and curves. For example, M 0,5 is isomorphic to the blow-up of P 2 in 4 points, and boundary divisors are the ten (−1)-curves. They generate Eff( M 0,5 )= NE 1 ( M 0,5 ). For n = 6, Keel (unpublished) and Vermeire [V] showed that Eff( M 0,6 ) is not generated by classes of boundary divisors. A new divisor has many interesting geometric interpretations but the known proof of its extremality is “numerical” rather than geometric in flavor, namely one computes its class and shows that it can not be written as a nontrivial sum of pseudoeffective divisors. To the best of our knowledge, this was the only known extremal divisor on M 0,n different from boundary divisors (of course one can also take pull-backs of the Keel–Vermeire divisor for various forgetful maps M 0,n → M 0,6 ). 1 arXiv:0809.1699v1 [math.AG] 10 Sep 2008