${\mathbb{G}}_m$-Equivariant Degenerations of del Pezzo Surfaces

Abstract

We study special ${\mathbb{G}}_m$-equivariant degenerations of a smooth del Pezzo surface $X$ induced by valuations that are log canonical places of $(X,C)$ for a nodal anti-canonical curve $C$. We show that the space of special valuations in the dual complex of $(X,C)$ is connected and admits a locally finite partition into sub-intervals, each associated to a ${\mathbb{G}}_m$-equivariant degeneration of $X$. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Kollár valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.

37 pages