We prove that the Cox ring of the moduli space M 0,6 , of stable rational curves with 6 marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are pull-backs of the hyperelliptic locus in M 3 via morphisms ρ : M 0,6 → M 3 that send a 6-pointed rational curve to a curve with 3 nodes by identifying 3 pairs of points. In particular this gives a self-contained proof of Hassett and Tschinkel's result about the effective cone of M 0,6 being

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... by the above mentioned divisors. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3852 ANA-MARIA CASTRAVET cone are both polyhedral and there are finitely many small modifications of X (i.e., varieties X isomorphic in codimension one to X) such that any moving divisor on X (i.e., a divisor whose base locus has codimension at least 2) is nef on one of the varieties X (see [HK] for the precise statements). It has been conjectured by Hu and Keel [HK] that any log-Fano variety has a finitely generated Cox ring. This has been recently proved in the groundbreaking paper [BCHM]. In [HK] Hu and Keel ask the following question: Question 1.1. Is the Cox ring of M 0,n finitely generated? As pointed out in [KM], the moduli space M 0,n is log-Fano only for n ≤ 6. We answer Question 1.1 for n = 6 by finding explicit generators. Our hope is that our method for finding generators, which proved to be useful in other circumstances (see [CT]), will eventually help answer Question 1.1 for larger n as well. Consider the Kapranov description of the moduli space M = M 0,6 . If p 1 , . . . , p 5 are points in linearly general position in P 3 , then M is the iterated blow-up of P 3 along p 1 , . If S ⊂ {1, . . . , 6} and |S| = 2, or 3, let ∆ S be the boundary divisor in M with general element a curve with two irreducible components with the partition of the markings given by S ∪ S c . In the Kapranov description, the boundary divisors ∆ S have the following classes: Notation 1.3. Let Q (ij)(kl) be the class of the proper transform of the unique quadric that contains all the points p 1 , . . . , p 5 and the lines l ik , l il , l jk , l jl : The divisor classes Q (ij)(kl) are exactly the divisors considered by Keel and Vermeire: for example, if one considers the map M 0,6 → M 3 given by identifying the pairs of points (12)(34)(56), then the class of the pull-back of the hyperelliptic locus in M 3 is computed in [HT] to be the class of Q (12)(34) . We call the divisors Q (ij)(kl) the Keel-Vermeire divisors. We prove the following: Theorem 1.4. The Cox ring of M 0,6 is generated by the sections (unique up to scaling) corresponding to the boundary divisors (i.e., Λ ijk and the exceptional divisors E i and E ij ) and the Keel-Vermeire divisors Q (ij)(kl) . The paper is divided as follows: Section 2 explains the strategy of proof; there are two main cases, the details of each are given in Section 3, respectively Section License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE COX RING OF M 0,6 3853 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE COX RING OF M 0,6 3855 Case I: Assume that D.L x ≥ 0, D.L y ≥ 0, D.L z ≥ 0. Notation 2.8. Denote by s ij the section on E 5 corresponding to the proper transform of the line l ij in P 2 . Let s i (i = 1, . . . , 4) be the sections corresponding to the exceptional divisors E i . Definition 2.9. We call a section s ∈ H 0 (E 5 , D) a distinguished section on E 5 if s can be written as a monomial in the sections s ij and s i . Since E 5 ∼ = M 0,5 is the blow-up of P 2 along q 1 , . . . , q 4 , by Lemma 7.3 the Cox ring Cox(E 5 ) of E 5 is generated by distinguished sections. The Main Claim follows from the following: Proposition 2.10. Under the assumptions of the Main Claim and the assumptions in Case I, the restriction map r E 5 : H 0 (M, D) → H 0 (E 5 , D) is surjective and one may lift any distinguished section (hence, any section) in H 0 (E 5 , D) to a section generated by distinguished sections in H 0 (M, D). The following is the main observation needed to prove Proposition 2.10: Main Observation -Case I. Distinguished sections on E 5 may be lifted to distinguished sections on M using the following rules: This is because Λ ij5 |E 5 = l ij , E i5|E 5 = E i (see Section 9, formula (9.3)). Sketch of Proof of Proposition 2.10. We lift a distinguished section s ∈ H 0 (E 5 , D) using the rules (2.2). Hence, there is a section t belonging to some H 0 (M, D ), where D = D and r E 5 (t ) = s. Notation 2.11. Let ∆ = D − D . Notation 2.12. Denote by X the iterated blow-up of P 3 in p 1 , . . . , p 4 and proper transforms of lines l ij (i, j ∈ {1, . . . , 4}).