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K-moduli space of log del pezzo surfaces
The concept of K-stability arises from the existence of Kähler-Einstein metrics on Fano manifolds in complex geometry. Recently,
algebraic geometers have used tools from K-stability theory to construct compact moduli spaces of Fano varieties; however, the
study of specific examples remains poorly understood. Even for certain surface pairs, the wall-crossing phenomenon remains unclear.
In this talk, I will first give a brief review of these theories. We will then apply the wall-crossing theory of K-moduli spaces to
study the K-moduli spaces of two classes of log del Pezzo surface pairs.
The main results consist of two parts. The first is the study of the K-moduli space of degree 8 del Pezzo surface pairs \((\mathbb{F}_1, C)\)
where \(C \in |−2K_{\mathbb{F}_1}|\). This moduli space is closely related to the moduli space of certain K3 surfaces with anti-symplectic involutions.
We use tools involving normalized volumes and T-singularity theory to determine all possible degenerations on the boundary. We then employ
the theory of complexity-one torus actions to calculate the positions of critical walls and classify critical surface pairs. Finally,
we describe the wall-crossing behavior of this K-moduli space via the local VGIT presentation. Notably, our moduli space provides
the first example of a K-moduli space that can be associated with non-reductive GIT.
The second part concerns the study of the moduli space of degree 9 del Pezzo surfaces with multiple boundaries \((\mathbb{P}^2; aQ + bL)\),
where \(Q\) is a plane quintic and \(L\) is a line. This provides the first example of higher-dimensional wall-crossing in K-moduli spaces.
This K-moduli space has a natural connection with the VGIT moduli space of degree 5 pairs investigated by Laza. We prove that our K-moduli
space is isomorphic to Laza’s moduli space when the coefficients of the surface pairs satisfy specific constraints.
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