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1. K-stability and birational models of moduli of quartic K3 surfaces

   https://doi.org/10.1007/s00222-022-01170-5  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (May 2023, Inventiones mathematicae)

   Abstract We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 . 

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2. The moduli spaces of sheaves on surfaces, pathologies and brill-noether problems

   https://doi.org/10.1007/978-3-319-94881-2_4  

   Cosun, I.; Huizenga, J. (November 2018, Abel symposia)

   Brill-Noether Theorems play a central role in the birational geometry of moduli spaces of sheaves on surfaces. This paper surveys recent work on the Brill-Noether problem for rational surfaces. In order to highlight some of the difficulties for more general surfaces, we show that moduli spaces of rank 2 sheaves on very general hypersurfaces of degree d in P3 can have arbitrarily many irreducible components as d tends to infinity. 

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3. Wall crossing for K‐moduli spaces of plane curves

   https://doi.org/10.1112/plms.12615  

   Ascher, Kenneth; DeVleming, Kristin; Liu, Yuchen (June 2024, Proceedings of the London Mathematical Society)

   Abstract We construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient 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. 

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4. K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

   https://doi.org/10.1017/S1474748021000384  

   ASCHER, KENNETH; DEVLEMING, KRISTIN; LIU, YUCHEN (September 2021, Journal of the Institute of Mathematics of Jussieu)

   Abstract 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. 

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5. Volumes And Random Matrices

   https://doi.org/10.1093/qmath/haaa035  

   Witten, Edward (February 2021, The Quarterly Journal of Mathematics)

   null (Ed.)

   Abstract This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020) 

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