K-moduli of Fano threefolds and genus four curves

Abstract In this article, we study the K-moduli space of Fano threefolds obtained by blowing up <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> \mathbb{P}^{3} along <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (2,3) -complete intersection curves. This K-moduli space is a two-step birational modification of the GIT moduli space of bidegree <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (3,3) -curves on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mn>1</m:mn> </m:msup> </m:mrow> </m:math> \mathbb{P}^{1}\times\mathbb{P}^{1} . As an application, we show that our K-moduli space appears as one model of the Hassett–Keel program for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mover accent="true"> <m:mi>M</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mn>4</m:mn> </m:msub> </m:math> \overline{M}_{4} . In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties. We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients.