Combinatorially minimal Mori dream surfaces of general type

In this paper, we suggest a new approach to study minimal surfaces of general type with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript g Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_g=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen [Mosc. Math. J. 8 (2008), pp. 711–757, 847]. First, we study general properties of combinatorially minimal Mori dream surfaces. Then we discuss how to apply these ideas to the study of minimal surfaces of general type with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript g Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_g=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are very important but still mysterious objects. In our previous paper [Adv. Math. 347 (2019), pp. 708-738], we provided several examples of Mori dream surfaces of general type with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript g Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_g = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and computed their effective cones explicitly. In this paper, we study their fibrations, explicit combinatorially minimal models and discuss singularities of the combinatorially minimal models. We also show that many minimal surfaces of general type with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript g Baseline equals 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_g=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> arise from the minimal resolutions of combinatorially minimal Mori dream surfaces.