Kollár showed that small deformations of elliptically fibered smooth K-torsion varieties with H^2(X,𝒪_X)=0 remain elliptically fibered. We extend this result to any fibered smooth K-torsion variety X with H^2(X,𝒪_X)=0, using Hodge theoretic techniques and the T^1-lifting criterion of Kawamata–Ran. More generally, our strategy implies that even without the cohomological assumption, small deformations of a semiample line bundle on a smooth K-torsion variety remain semiample up to homological equivalence.

This survey article is an accompaniment to the 2025 Summer Research Institute in Algebraic Geometry Bootcamp on K-stability and K-moduli. It is aimed at graduate students and intended to provide the necessary background to begin research on explicit K-moduli problems.

There have been major developments in the theory of moduli of varieties in the past decade, essentially settling the construction of moduli spaces of log canonically polarized slc pairs and moduli spaces of K-polystable log Fano pairs. Given the construction of these moduli spaces of pairs (X, D), it is natural to ask how the moduli spaces vary as the coefficients of D are perturbed. This phenomenon is known as wall crossing, the theory of which has been developed in several important cases in the past five years. This semi-expository article is an introduction to moduli of varieties and wall crossing, capturing a portion of the theory developed in the past several years. It also introduces tools and techniques used in explicit computations and examples, applying them in new examples.

We prove the K-polystability of the general Fano complete intersection of arbitrary multidegree and dimension, and the K-stability of the general Fano complete intersection that is not isomorphic to projective space or a quadric hypersurface. We prove analogous results for certain smooth weighted complete intersections.

The Hassett–Keel program seeks to give a modular interpretation to the steps of the log minimal model program of ℳ_g. The goal of this paper is to complete the Hassett–Keel program in genus four, supplementing earlier results of Casalaina-Martin–Jensen–Laza and Alper–Fedorchuk–Smyth–van der Wyck. The main tools we use are wall crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi–Yau surface pairs. We also give a construction of the hyperelliptic flip using a stackified Chow quotient which is expected to generalize to higher genus.

The Hassett--Keel program seeks to give a modular interpretation to the steps of the log minimal model program of $\overline{\mathcal{M}}_g$. The goal of this paper is to complete the Hassett--Keel program in genus four, supplementing earlier results of Casalaina-Martin--Jensen--Laza and Alper--Fedorchuk--Smyth--van der Wyck. The main tools we use are wall crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi--Yau surface pairs. We also give a construction of the hyperelliptic flip using a stackified Chow quotient which is expected to generalize to higher genus.

We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-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 $d \geq 4$ as boundary divisors of $\mathbb{P}^2$. 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.

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2.18. These Fano threefolds are double covers of $\mathbb P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this family is K-stable. A member of family No 2.18 admits the structures of a conic bundle and a quadric surface bundle. We prove that K-polystable limits of these Fano threefolds admit conic bundle structures, but not necessarily del Pezzo fibration structures. We study this K-moduli space via the moduli space of log Fano pairs $(\mathbb P^1\times\mathbb P^2, c R)$ for $c=1/2$ and $R$ a $(2,2)$-divisor, which we construct using wall-crossings. In the case where the divisor is proportional to the anti-canonical divisor, the first author, together with Ascher and Liu, developed a framework for wall crossings in K-moduli and proved that there are only finitely many walls, which occur at rational values of the coefficient $c$. This paper constructs the first example of wall-crossing in K-moduli in the non-proportional setting, and we find a wall at an irrational value of $c$. In particular, we obtain explicit descriptions of the GIT and K-moduli spaces (for $c \leq 1/2$) of these $(2,2)$-divisors. Furthermore, using the conic bundle structure, we study the relationship with the GIT moduli space of plane quartic curves.

For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of quintic curves that give counterexamples in dimension 1 (Horikawa constructed similar examples for quintics in dimension 2). In this paper we propose a conjecture explaining these examples using Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane. In particular, if p is a prime number that is not a Markov number, we conjecture that any smooth projective limit of plane curves of degree p is a plane curve. The main results are to prove this conjecture for degree 7 curves and to extend Griffin's counterexamples to all prime numbers that are also Markov numbers.

We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and composite degree in the classification and study the relationship between prime or composite degree and the number of distinct topological types of cuspidal singularities. Motivated by wall-crossing in moduli of curves, we also survey several geometric properties of rational unicuspidal plane curves.

We develop the moduli theory of boundary polarized CY pairs, which are slc Calabi-Yau pairs $(X,D)$ such that $D$ is ample. The motivation for studying this moduli problem is to construct a moduli space at the Calabi-Yau wall interpolating between certain K-moduli and KSBA moduli spaces. We prove that the moduli stack of boundary polarized CY pairs is S-complete, $\Theta$-reductive, and satisfies the existence part of the valuative criterion for properness, which are steps towards constructing a proper moduli space. A key obstacle in this theory is that the irreducible components of the moduli stack are not in general of finite type. Despite this issue, in the case of pairs $(X,D)$ where $X$ is a degeneration of $\mathbb{P}^2$, we construct a projective moduli space on which the Hodge line bundle is ample. As a consequence, we complete the proof of a conjecture of Prokhorov and Shokurov in relative dimension two.

We show that the K-moduli spaces of log Fano pairs (P-1 x 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 and O'Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of (4,4) curves on P-1 x P-1 and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

The main goal of this paper is to construct a compactification of the moduli space of degree $d \geqslant 5$ surfaces in $\mathbb {P}^{3}_{{{\mathbb {C}}}}$ , i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb {P}^{3}$ and whose boundary points correspond to degenerations of such surfaces. 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 $Z = {{\mathbb {P}}}^{3}$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.

We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ 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 $${\mathbb {P}}^3$$ .

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.

Motivated by the study of rationally connected fibrations, we study different notions of birationally simple fibrations. Our main result is the construction of maximal Chow constant and cohomologically constant fibrations. This paper is largely self-contained and we prove a number of basic properties of these fibrations. One application is to the classification of "rationalizations of singularities of cones." We also consider consequences for the Chow groups of the generic fiber of a Chow constant fibration.

Compact Moduli of Surfaces in Three-Dimensional Projective Space

We show that all subvarieties of a quasi-projective variety with positive log cotangent bundle are of log general type. In addition, we show that smooth quasi-projective varieties with positive and globally generated log cotangent have finitely many integral points, generalizing a theorem of Moriwaki. Finally, we prove 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 positive log cotangent sheaf are uniformly bounded.