We study semistable extremal 3-fold neighborhoods, which are fundamental building blocks of birational geometry, following earlier work of Mori, Kollár, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general k1A neighborhoods. The novelty of our approach is to show that k1A belong to the same deformation family as k2A, in fact we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which can be interpreted as a sequence of mutations in the cluster algebra. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Kollár-Shepherd-Barron-Alexeev boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero. s 1 In some papers C is allowed to be reducible, however we can always reduce to the irreducible case by factoring f locally analytically over Q ∈ Y, see [K88, 8.4].2 This is a special case of the general elephant conjecture of Miles Reid. 2 6 Acknowledgements. . We would like to thank I. Dolgachev, J. Kollár, S. Mori, and M. Reid for helpful discussions. In particular, M. Reid explained to us in detail his joint work with G. Brown which describes explicitly the graded ring of a k2A flip [BR13].Contents m 2 1(1, m 1 a 1 − 1)) for some m 0 , m 2 ∈ N such that m 0 ≡ m 2 (m 1 a 1 − 1) mod m 2 1 . 9

We prove a strong relation between Chern and log Chern invariants of algebraic surfaces. For a given arrangement of curves, we find nonsingular projective surfaces with Chern ratio arbitrarily close to the log Chern ratio of the log surface defined by the arrangement. Our method is based on sequences of random p-th root covers, which exploit a certain large scale behavior of Dedekind sums and lengths of continued fractions. We show that randomness is necessary for our asymptotic result, providing another instance of "randomness implies optimal". As an application over C, we construct nonsingular simply connected projective surfaces of general type with large Chern ratio. In particular, we improve the Persson-Peters-Xiao record for Chern ratios of such surfaces. Contents

We explicitly bound T-singularities on normal projective surfaces W with one singularity, and KW ample. This bound depends only on K 2 W , and it is optimal when W is not rational. We classify and realize surfaces attaining the bound for each Kodaira dimension of the minimal resolution of W . This answers effectiveness of bounds (see [A94], [AM04], [L99]) for those surfaces.

We show a one-to-one correspondence between arrangements of d lines in P 2 , and lines in P d−2 . We apply this correspondence to classify (3, q)-nets over C for all q ≤ 6. When q = 6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable over C. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3, 8)-nets corresponding to the Quaternion group.

Abstract. We prove that for any number r ∈ [2, 3], there are spin (resp. non-spin and minimal) simply connected complex surfaces of general type X with c 2 1 (X)/c 2 (X) arbitrarily close to r. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any r ∈ [1, 3] and any integer q ≥ 0, there are minimal complex surfaces of general type X with c 2 1 (X)/c 2 (X) arbitrarily close to r, and π 1 (X) isomorphic to the fundamental group of a compact Riemann surface of genus q.