By work of Gallardo–Kerr–Schaffler, it is known that Naruki’s compactification of the moduli space of marked cubic surfaces is isomorphic to the normalization of the Kollár, Shepherd-Barron, and Alexeev (KSBA) compactification parametrizing pairs (S,(1/9 + ε)D), with D the sum of the 27 marked lines on S, and their stable degenerations. In the current paper, we show that the normalization assumption is not necessary as we prove that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is done by studying the automorphisms and the Q-Gorenstein obstructions of the stable pairs parametrized by it.
Fang, H., Schaffler, L., Wu, X. (2025). FINENESS AND SMOOTHNESS OF A KSBA MODULI OF MARKED CUBIC SURFACES. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 153(12), 5133-5146 [10.1090/proc/17404].
FINENESS AND SMOOTHNESS OF A KSBA MODULI OF MARKED CUBIC SURFACES
Schaffler L.
;
2025-01-01
Abstract
By work of Gallardo–Kerr–Schaffler, it is known that Naruki’s compactification of the moduli space of marked cubic surfaces is isomorphic to the normalization of the Kollár, Shepherd-Barron, and Alexeev (KSBA) compactification parametrizing pairs (S,(1/9 + ε)D), with D the sum of the 27 marked lines on S, and their stable degenerations. In the current paper, we show that the normalization assumption is not necessary as we prove that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is done by studying the automorphisms and the Q-Gorenstein obstructions of the stable pairs parametrized by it.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
