Wall crossing for K-moduli spaces of plane curves

Abstract

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 ...

On the volume of some Fano K-moduli spaces

“…Using the outstanding effort of Guenancia and Paun [28] in understanding the global Monge Ampere equation in the conic Kähler-Einstein case, and the effort of Berman in [26], and lately by Tian and Wang in [29] on understanding the YTD conjecture in the simple normal crossing log case we are able to define the Weil-Peterson metric for the smooth locus of K-polystable families of log Fano pairs along the line of Fujiki and Schumacher in [30]. Then, generalising the results given in [37] we are able to extend it to the whole family, and with the same arguments of [37] and [41] we are also able to extend it to the K-moduli space. More technically, in Theorem 2.3 we prove that for a family f : (X , D) → B of log K-polystable Fano varieties, there exists a continuous Deligne's Pairing metric h DP on the log CM line bundle λ CM,D such that the Weil-Petersson metric ω WP extends as a positive current to the whole base B.…”

On K-stability of some del Pezzo surfaces of Fano index 2

“…Next, we treat the "only if" part. In fact, this follows from moduli comparison arguments as in [ADL19]. Let A := H 0 (P 1 , O P 1 (2a)) be the affine space parametrizing degree 2a binary forms.…”

“…Every toric variety we consider is normal. We do not even try to write down the definitions of K-(poly/semi)stability of Fano varieties and of log Fano pairs: we refer the reader to the excellent survey [Xu20a], the paper [ADL19], and to the references therein.…”

On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line

“…Where λ CM is the CM line bundle defined in Chapter two and λ CH is the Chow line bundle, defined as the leading order term of the Hilbert Mumford expansion, namely λ CH = λ n+1 , and λ CH,D = k i=1 λ di CH . We outline some easy and well known properties of the log CM line bundle, for a full description of the below facts see [91].…”