Degenerations of cluster type varieties

Abstract

We study degenerations of cluster type varieties and pairs. Our first theorem proves that degenerations of toric pairs are finite quotients of toric pairs. In a similar vein, under some mild conditions, we prove that degenerations of cluster type pairs are finite quotients of cluster type pairs. Then, we focus on degenerations of cluster type surfaces. We give some general criteria for the existence of $1$-complements on degenerations of toric surfaces. We prove that for almost all $(a,b,c)\in \mathbb{Z}_{\geq 1}^3$ the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial degenerations. In particular, for a Markov triple $(a,b,c)\in \mathbb{Z}_{\geq 2}^3$, we prove that $\mathbb{P}(a^2,b^2,c^2)$ admits no non-trivial degenerations. Finally, we give a complete classification of the degenerations of $\mathbb{P}(1,1,n)$ for $n\geq 3$.