JongHae Keum

In this paper, we suggest a new approach to study minimal surfaces of general type with p g = 0 p_g=0 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 p g = 0 p_g=0 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 p g = 0 p_g = 0 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 p g = 0 p_g=0 arise from the minimal resolutions of combinatorially minimal Mori dream surfaces.

We show that Shavel-type surfaces are fake ‫-ޑ‬homology quadrics of even type which are not Mori dream surfaces, yet there are infinitely many primes p such that the reduction modulo p is a Mori dream surface. We investigate fake ‫-ޑ‬homology quadrics, first concerning the property of being a Mori dream surface, then trying to determine which families are of even type among the surfaces isogenous to a higher product which are fake ‫-ޑ‬homology quadrics.

We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism.

We show, for several fake projective planes with a nontrivial group of automorphisms, that the bicanonical map is an embedding.

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding.

If an automorphism of a projective K3 surface with Picard number 2 is of infinite order, then the automorphism corresponds to a solution of Pell equation. In this paper, by solving this equation, we determine all Salem polynomials of symplectic and anti-symplectic automorphisms of projective K3 surfaces with Picard number 2.