We construct proper good moduli spaces parametrizing K-polystable Q-Gorenstein smoothable log Fano pairs (X, cD), where X is a Fano variety and D is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as c varies. The main application in this paper is the case of plane curves of degree d ≥ 4 as boundary divisors of P 2 . In this case, we show that when the coefficient c is small, the K-moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K-moduli spaces are weighted blow-ups of Kirwan type. We also describe all wall crossings for degree 4, 5, 6 and relate the final K-moduli spaces to Hacking's compactification and the moduli of K3 surfaces. Patakfalvi, Sam Payne, Chenyang Xu, Qizheng Yin, and Ziquan Zhuang. We wish to thank Valery Alexeev, Yongnam Lee, Yuji Odaka, and Hendrik Süß for many useful comments on a preprint. We are especially grateful to Xiaowei Wang for fruitful discussions regarding technical issues in Section 3, Feng Wang for kindly providing a proof of Theorem 3.6, and Quentin Posva for providing us a draft of [Pos19].We thank Patricio Gallardo for several helpful correspondences regarding GIT for plane quintics and Kirwan desingularizations of GIT quotients during the beginning stages of this project. We also note that Theorem 1.3 was proven independently of the related [GMGS18, Theorem 1.2].Parts of this paper were completed while the authors were in residence at MSRI in Spring 2019. KA was supported in part by an NSF Postdoctoral Fellowship, and would like to thank the math department of University of Washington for providing wonderful visiting conditions. KD was partially supported by the Gamelin Endowed Postdoctoral Fellowship of the MSRI (NSF No. DMS-1440140). YL was partially supported by the Della Pietra Endowed Postdoctoral Fellowship of the MSRI (NSF No. DMS-1440140).Finally, we thank Sándor Kovács and Karl Schwede for organizing a special session at the AMS Sectional at Portland State University in Apr. 2018 where this collaboration began. 6.2. Relating degree 4 and 6 plane curves to K3 surfaces 48 7. The second wall crossing for plane quintics 50 7.1. K-polystable replacement of A 12 -quintic curve 50 7.2. Proof of second wall crossing 52 7.3. Applications to higher degree 55 8. Log Fano wall crossings for K-moduli spaces of plane quintics 56 8.1. GIT of plane quintics 56 8.2. Explicit wall crossings 57 8.3. Proofs 58 9. Projectivity, birational contractions, and the log Calabi-Yau wall crossing 61 9.1. Projectivity 61 9.2. Birational contractions along wall crossings 62 9.3. Quartics and sextics revisited 66 9.4. Log Calabi-Yau quintics 68 9.5. Higher dimensional applications 71 Appendix A. Calculations of K-semistable thresholds and K-polystable replacements 72 A.1. A 12 73 A.2. A 11 77 A.3. A 10 82 A.4. Valuative criterion computations 85 References 87 2. Preliminaries 2.1. K-stability of log Fano pairs. In this section, we give a review of K-stability ...

We show that the K-moduli spaces of log Fano pairs (P 1 ×P 1 , cC) where C is a (4, 4)curve and their wall crossings coincide with the VGIT quotients of (2, 4) complete intersection curves in P 3 . This, together with recent results by Laza-O'Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of (4, 4)-curves on P 1 × P 1 and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces. Contents 1. Introduction 1 2. Preliminaries 4 3. Overview of previous results, Laza-O'Grady, and VGIT 9 4. Degenerations of P 1 × P 1 in K-moduli spaces 13 5. Wall crossings for K-moduli and GIT 18 6. Some results for (d, d) curves 26 References 29

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang–Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.

The main goal of this paper is to construct a compactification of the moduli space of degree d ≥ 5 surfaces in P 3 C , i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in P 3 and whose boundary points correspond to degenerations of such surfaces. Motivated by numerous others (see, for example [KSB88], [Ale96], [Hac04]), we consider a divisor D on a Fano variety Z as a pair (Z, D) satisfying certain properties. We find a modular compactification of such pairs and, in the case of Z = P 3 and D a surface, use their properties to classify the pairs on the boundary of the moduli space.