Fineness and smoothness of a KSBA moduli of marked cubic surfaces

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma left-parenthesis one ninth plus epsilon right-parenthesis upper D right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>9</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi> ε </mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left (S,\left (\frac {1}{9}+\varepsilon \right )D\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the sum of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="27"> <mml:semantics> <mml:mn>27</mml:mn> <mml:annotation encoding="application/x-tex">27</mml:annotation> </mml:semantics> </mml:math> </inline-formula> marked lines on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Gorenstein obstructions of the stable pairs parametrized by it.