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

Using K-moduli spaces for log quasimaps $q_t\text{\hspace{0.17em}⁣}:({\mathbb{P}}^1,\frac{1-t}{12}D)\to [{\mathbb{A}}^2\mathrm{/}{\mathbb{G}}_m]$ of degree twelve with twelve points and weight $t\mathrm{/}12$, 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 $${\mathcal{M}}_t\cong \mathrm{Proj}R\text{\hspace{0.17em}⁣}(X_o,\mathcal{L}+\frac{t}{2}\mathrm{\Delta }(6)+\frac{t}{3}\mathrm{\Delta }(9)),$$ where $\mathcal{L}$ is the automorphic $\mathbb{Q}$-line bundle and $\mathrm{\Delta }(6),\mathrm{\Delta }(9)$ 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 $t\in (0,1\mathrm{/}7)$ is Looijenga’s semi-toroidal compactification.