Jie Liu

TBA

A question that turns out to be extremely entertaining and instructive concern the graded ring over an orbifold curve \(C\) of genus 2 marked with distinct points \(P + Q\) in \(|K_C|\), and polarized by the fractional divisor \(\frac{1}{2}P + \frac{3}{5}Q\). Theorem 1 says that \(C\) is the weighted plane curve \(C(11)\) in \(\mathbb{P}(1,2,5)\). The calculation provides an opportunity to explain many points around orbifold RR, that are elementary but deserve to be better known.
Theorem 1 forms base camp for an ascent to one interesting case of a K3 surface and a Fano 3-fold of codimension 5. GRDB list more than 50,000 candidate Hilbert series of Fano 3-folds, of which only around 1,000 have been seriously studied and partly understood. Current joint work with SUZUKI Kaori successfully constructs a case of #41058, and corrects the GRDB entry for #41245.

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.