No abstract uploaded!

We develop a new technique to study ranks of multiplication maps for linear series via limit linear series and degenerations to chains of elliptic curves. We prove an elementary criterion and apply it to proving cases of the Maximal Rank Conjecture. We give a new proof of the case of quadrics, and also treat several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.

No abstract uploaded!

No abstract uploaded!

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.