Degenerations of cluster type varieties

TL;DR

This paper investigates how cluster type varieties and pairs degenerate, establishing finiteness results, criteria for degenerations of surfaces, and classifying degenerations of certain weighted projective planes.

Contribution

It proves that degenerations of toric and cluster pairs are finite quotients, and classifies degenerations of specific weighted projective planes, advancing understanding of their degeneration behavior.

Findings

Degenerations of toric pairs are finite quotients of toric pairs.

Degenerations of cluster pairs are finite quotients under mild conditions.

Weighted projective planes $P(a,b,c)$ with almost all $(a,b,c)$ have no non-trivial degenerations.

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 $n\geq 3$.