In this paper we construct a simply connected, minimal, complex surface of general type with pg = 0 and K 2 = 2 using a rational blow-down surgery and a Q-Gorenstein smoothing theory.In this section we develop a theory of Q-Gorenstein smoothing for projective surfaces with special quotient singularities, which is a key technical ingredient in our result.Definition. Let X be a normal projective surface with quotient singularities. Let X → ∆ (or X /∆) be a flat family of projective surfaces over a small disk ∆. The one-parameter family of surfaces X → ∆ is called a Q-Gorenstein smoothing of X if it satisfies the following three conditions; (i) the general fiber X t is a smooth projective surface,

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple αδ of the divisor δ of singular curves as the boundary divisor, construct the log canonical model for the pair (M 3 , αδ) using geometric invariant theory as we vary α from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of M 3 , some new and others classical, while the exception gives a new modular compactification of M 3 . ′ 4 31 4.3. P 4 and P ′ 4 as log canonical models 32 4.4. Kondo's compact moduli space 33 References 34 1 M 3 (1) ≃ M 3 T M 3 ( 9 11 ) ≃ M ps 3 Ψ M 3 (7/10) ≃ M cs 3 M 3 ( 17 28 ) ≃ Q * This program was initiated by Brendan Hassett and Sean Keel, and the ideas were further developed in [HH06]. The genus two case was completely worked out by Hassett in [Has05] (see also [HL07]) and the first couple of steps of the program for the higher genera case were completed in [HH06] and [HH07].We work over an algebraically closed field k of characteristic zero. Acknowledgement. D.H. would like to thank Brendan Hassett for suggesting this problem and for many helpful conversations. He was partially supported by KIAS. He gratefully acknowledges Bumsig Kim for the hospitality and for useful conversations.Y.L. would like to thank Shigeyuki Kondo for his explanation of his compact moduli space, Shigefumi Mori for his explanation of birational geometry, Shigeru Mukai for his valuable comments on the geometric invariant theory.

In this paper we study birational immersions from a very general smooth plane curve to a non-rational surface with pg = q = 0 to treat dominant rational maps from a very general surface X of degree≥ 5 in P 3 to smooth projective surfaces Y . Based on the classification theory of algebraic surfaces, Hodge theory, and deformation theory, we prove that there is no dominant rational map from X to Y unless Y is rational or Y is birational to X.

We completely classify tri-canonically embedded curves of genus two that are Chow semistable, and identify the moduli space of them with the compact moduli space of binary sextics. This moduli space is the log canonical model for the pair M 2 , α 0 + 1+α 2 1 + 1 2 for 7/10 < α ≤ 9/11 whose log canonical divisor pulls back to K M 2 + αδ on the moduli stack.

Algebraically simply connected surfaces of general type with pg = q = 0 and 1 K 2 4 in positive characteristic (with one exception in K 2 = 4) are presented by using a Q-Gorenstein smoothing of two-dimensional toric singularities, a generalization of Lee-Park's construction [36] to the positive characteristic case, and Grothendieck's specialization theorem for the fundamental group.