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In this paper, we suggest a new approach to study minimal surfaces of general type with $p_g=0$ via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen. 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 $p_g=0$ which are very important but still mysterious objects. In our previous paper, we provided several examples of Mori dream surfaces of general type with $p_g=0$ 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 $p_g=0$ arise from the minimal resolutions of combinatorially minimal Mori dream surfaces.

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., Comment: 12 printed pages, typos are corrected and make some minor modifications

In this paper we study effective, nef and semiample cones of minimal surfaces of general type with $p_g=0.$ We provide examples of minimal surfaces of general type with $p_g=0, 2 \leq K^2 \leq 9$ which are Mori dream spaces. On these examples we also give explicit description of effective cones and all irreducible reduced curves of negative self-intersection. We also present non-minimal surfaces of general type with $p_g=0$ that are not Mori dream surfaces., Comment: Add examples of non-minimal surfaces of general type with $p_g=0$ that are not Mori dream surfaces. Fix a few typos and add more details

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., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience

We show, for several fake projective planes with nontrivial automorphism group, that the bicanonical map is an embedding., Comment: A new section (Section 6) is added to point out that, due to many wrong proofs, the main result of the paper by S.-K. Yeung [Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients, Forum Math. 2017;30(2): 419-432] is not proven at all

We prove that the bicanonical map of the Cartwright-Steger surface is an embedding. We also discuss two minimal surfaces of general type, both covered by the Cartwright-Steger surface. One has $K^2=2$, $p_g=1$, $\pi_1=\{1\}$ and the other has $K^2=1$, $p_g=0$, $\pi_1=\mathbb{Z}/2\mathbb{Z}$., Comment: 15 pages. Stronger result than in the previous version

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.

In this short note we announce explicit equations of a fake projective plane in its bicanonical embedding in $\mathbb C\mathbb P^9$., Comment: 2 pages + 2 pages of tables

Let $X$ be a projective surface or a hyperk\"ahler manifold and $G \le Aut(X)$. We give a necessary and sufficient condition for the existence of a non-trivial $G$-equivariant fibration on $X$. We also show that two automorphisms $g_i$ of positive entropy and polarized by the same nef divisor are the same up to powers, provided that either $X$ is not an abelian surface or the $g_i$ share at least one common periodic point. The surface case is known among experts, but we treat this case together with the hyperk\"ahler case using the same language of hyperbolic lattice and following Ratcliffe or Oguiso. This arXiv version contains proofs omitted in the print version., Comment: Journal of the London Mathematical Society (to appear), 16 pages. This is the final arXiv version. The printed version has some proofs omitted as suggested by the referees

We present the complete list of all singularity types on Gorenstein $\mathbb{Q}$-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of $58$ possible singularity types, each except two types supported by an example., Comment: Added in Proof. The type $A_3 \oplus 3A_2$ realizes on the Enriques surface constructed by Slawomir Rams and Matthias Sch\"utt [On Enriques surfaces with four cusps, arXiv:1404.3924, Example 3.9]

In any characteristic different from 2 and 5, Kond\=o gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic $p\neq 2, 5$, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kond\=o., Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:1402.6803, arXiv:1204.1711

For every fake projective plane $X$ with automorphism group of order 21, we prove that $H^i(X, 2L)=0$ for all $i$ and for every ample line bundle $L$ with $L^2=1$. For every fake projective plane with automorphism group of order 9, we prove the same vanishing for every cubic root (and its twist by a 2-torsion) of the canonical bundle $K$. As an immediate consequence, there are exceptional sequences of length 3 on such fake projective planes., Comment: 17 pages

In each characteristic $p\neq 2, 3$, it was shown in a previous work that the order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it is shown that in each characteristic $p\neq 2, 3$ a K3 surface with a cyclic action of order 66 is unique up to isomorphism. The equation of the unique surface is given explicitly in the tame case ($p\nmid 66$) and in the wild case ($p=11$)., Comment: 12 papges. arXiv admin note: text overlap with arXiv:1204.1711

In characteristic $p=0$ or $p>5$, we show that a K3 surface with an order 60 automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular K3 surface with Artin invariant 1 in characteristic $p=11$ (mod 12) by a cyclic symmetry of order 60., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1203.5616

We determine all posible orders of automorphisms of finite order of complex K3 surfaces or of K3 surfaces in characteristic $p>3$. E.g., a positive integer $N$ is the order of an automorphism of a complex K3 surface if and only if $\phi(N)\le 20$ where $\phi$ is the Euler function. In particular, 66 is the maximum finite order in each characteristic $p\neq 2,3$. As a consequence, we give a bound for the orders of finite groups acting on K3 surfaces in characteristic $p>7$., Comment: 43 pages

We determine all possible orders of automorphisms of complex K3 surfaces. A positive integer N is the order of an automorphism of a complex K3 surface if and only if $\phi(N) \leq 20$ where $\phi$ is the Euler function., Comment: This paper has been included in the author's another paper arXiv;1203.5616

We give a criterion for a projective surface to become a quotient of a fake projective plane. We also give a detailed information on the elliptic fibration of a $(2,3)$-elliptic surface that is the minimal resolution of a quotient of a fake projective plane. As a consequence, we give a classification of ${\mathbb Q}$-homology projective planes with cusps only., Comment: 14 pages. Minor changes in Theorem 0.6 and Remark 0.7 were made

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new examples of a minimal surface of general type with $p_g=0$, $\pi_1=\mathbb{Z}/2\mathbb{Z}$, and $K^2\le 4$., Comment: This paper combines two preprints arXiv:1008.1222 and arXiv:0910.3476. The results are slightly improved

Koll\'ar gave a series of examples of rational surfaces of Picard number $1$ with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples have the same singularities as Koll\'ar's examples., Comment: 13 pages, 7 figures

The paper is a generalization of a result of I. Dolgachev, M. Mendes Lopes, and R. Pardini. We prove that a smooth projective complex surface $X$, not necessarily minimal, contains $h^{1,1}(X)-1$ disjoint $(-2)$-curves if and only if $X$ is isomorphic to a minimal rational ruled surface ${\bf F}_2$ or ${\bf P}^2$ or a fake projective plane. We also describe smooth projective complex surfaces $X$ with $h^{1,1}(X)-2$ disjoint $(-2)$-curves., Comment: 9 pages. A missing case was added to Theorem 1.4. Minor changes were made

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Koll\'ar formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with the second Betti number $b_2(S) = 1$ and with quotient singularities only has at most 3 singular points if its smooth locus $S^0$ is simply-connected. In a previous paper, we have confirmed the conjecture when $S$ has at least one non-cyclic quotient singularity. In this paper, we prove the conjecture either when $S$ is not rational or when $-K_S$ is ample., Comment: 41 pages. The proof of Proposition 4.2 and Section 8 are simplified

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere ${\mathbb S}^5$ has at most 3 non-free orbits. Using a certain one-to-one correspondence, Koll\'ar formulated the algebraic version of the Montgomery-Yang problem: every projective surface $S$ with quotient singularities such that $b_2(S) = 1$ has at most 3 singular points if its smooth locus $S^0$ is simply-connected. In this paper, we prove the conjecture under the assumption that $S$ has at least one noncyclic singularity. In the course of the proof, we classify projective surfaces $S$ with quotient singularities such that (i) $b_2(S) = 1$, (ii) $H_1(S^0, \mathbb{Z}) = 0$, and (iii) $S$ has 4 or more singular points, not all cyclic, and prove that all such surfaces have $\pi_1(S^0)\cong \mathfrak{A}_5$, the icosahedral group., Comment: 26 pages. A statement in Theorem 1.5(4) is corrected, hence the corresponding statements in Proposition 3.5 and Example 3.6 are also corrected

Recently, Prasad and Yeung classified all possible fundamental groups of fake projective planes. According to their result, many fake projective planes admit a nontrivial group of automorphisms, and in that case it is isomorphic to $\bbZ/3\bbZ$, $\bbZ/7\bbZ$, $7:3$, or $(\bbZ/3\bbZ)^2$, where $7:3$ is the unique non-abelian group of order 21. Let $G$ be a group of automorphisms of a fake projective plane $X$. In this paper we classify all possible structures of the quotient surface $X/G$ and its minimal resolution., Comment: 16 pages, with minor change of the exposition

A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane $S$ with quotient singularities such that $K_S$ is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type $ 3A_1 \oplus 2A_3$. We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case., Comment: 23 pages. changed the exposition of the previous version

First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we shall confirm the conjecture of Tits type for complex tori, hyperk\"ahler manifolds, surfaces, and minimal threefolds., Comment: 16 pages

We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group of order 11, $11\rtimes 5$, $L_2(11)$ and the Mathieu groups $M_{11}$, $M_{22}$. We also show that a surface $X$ admitting an automorphism $g$ of order 11 admits a $g$-invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces., Comment: 19 pages

Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected. We confirm the conjecture under the additional condition that the exceptional divisor in a minimal resolution of $S$ has at most 3 components over each singular point of $S$., Comment: An error in the previous version was corrected. To appear in the Proceedings of the Conference in honor of Igor Dolgachev on his 60th birthday

A fake projective plane is a compact complex manifold of dimension 2 which has the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by D. Mumford, there exists at least one such surface. In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface., Comment: An error in the previous version has been corrected. 9 pages

We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface., Comment: submitted to European Journal of Combinatorics (special issue on Groups and Geometry)

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic $p$ under the assumptions that (i) the order of the group is coprime to $p$ and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without the assumption (ii) we classify all possible new groups which may appear. We prove that the assumption on the order of the group is always satisfied if $p > 11$ and if $p=2,3,5,11$ we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by $p$ which are not contained in Mukai's list., Comment: The final version to appear in Annals of Mathematics. 42 pages

The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a complex algebraic geometry of $K3$ surfaces., Comment: 24 pages, 6 figures

We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus., Comment: AMS TeX

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16. The latter embeds naturally in the even unimodular lattice $II^{1,25}$ of rank 26 and signature $(1,25)$ as the orthogonal complement of a root sublattice of rank 10. Our generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of S. Kond$\bar {\roman o}$. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface., Comment: 25 pages, 2 figures