K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
Abstract
Using K-moduli spaces for log quasimaps qt:(ℙ1,(1−t)/12D)→[𝔸2/𝔾m] of degree twelve with twelve points and weight t/12, we construct a modular interpolation {ℳt}0≤ t≤1 between the Baily--Borel compactification of the Heckman--Looijenga ball quotient Xo and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational t∈[0,1], identify ℳt≅ProjR (Xo,ℒ+t/2Δ(6)+t/3Δ(9)), where ℒ is the automorphic ℚ-line bundle and $Δ(6),Δ(9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on Xo 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 ℳt for t∈ (0,1/7) is Looijenga's semi-toroidal compactification.
Related
- Log Calabi-Yau wall crossing for moduli of cubic surfaces
- On wall-crossing invariant combinations of Welschinger numbers for rational surfaces with a conic bundle
- Crepant resolutions of fiber products and Kummer fibrations of rational elliptic surfaces with non-reduced fibers
- Local Constancy of the Betti Automorphic Category over the Moduli of Curves
- Surfaces Associated to Zeros of Automorphic L-functions
BibTeX
@misc{hattori2026kmoduli,
title = {{K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces}},
author = {Masafumi Hattori and Yota Maeda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10777},
eprint = {2610.10777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10777},
}
RIS
TY - UNPB
TI - K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
AU - Masafumi Hattori
AU - Yota Maeda
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10777
UR - https://arxiv.org/abs/2610.10777
AB - Using K-moduli spaces for log quasimaps q_t:(ℙ^1,(1−t)/12D)→[𝔸^2/𝔾_m] of degree twelve with twelve points and weight t/12, we construct a modular interpolation {ℳ_t}_0≤ t≤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∈[0,1], identify ℳ_t≅ProjR (X_o,ℒ+t/2Δ(6)+t/3Δ(9)), where ℒ is the automorphic ℚ-line bundle and $Δ(6),Δ(9)$ 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 ℳ_t for t∈ (0,1/7) is Looijenga's semi-toroidal compactification.
ER -