K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces

Abstract

Using K-moduli spaces for log quasimaps $q_t:\left({\mathbb{P}}^1,\frac{1-t}{12}D\right)\to [{\mathbb{A}}^2/{\mathbb{G}}_m]$ of degree twelve with twelve points and weight $t/12$, we construct a modular interpolation $\{{\mathcal{M}}_t{\}}_{0\le t\le 1}$ between the Baily--Borel compactification of the Heckman--Looijenga ball quotient $X_o$ and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational $t\in [0,1]$, identify \[ \mathcal M_t\cong\operatorname{Proj}R\!\left(X_o,\mathcal L+\frac t2Δ(6)+\frac t3Δ(9)\right), \] where $\mathcal{L}$ is the automorphic $\mathbb{Q}$-line bundle and $\Delta (6),\Delta (9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on $X_o$ via a Borcherds product, whose divisor gives an independent relation among the Heegner divisors. This relation provides a key input for determining the birational transformations in the K-moduli wall-crossing. As part of this analysis, we show that the first positive chamber ${\mathcal{M}}_t$ for $t\in (0,1/7)$ is Looijenga's semi-toroidal compactification.

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