We prove that every irreducible component of the coarse Kollár-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi–Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of Hacking–Keel–Yu. The proof combines tools from log smooth deformation theory, the minimal model program, punctured log Gromov–Witten theory, and mirror symmetry.

We study the moduli space of constant scalar curvature Kähler (cscK) surfaces around toric surfaces. To this end, we introduce the class of foldable surfaces: smooth toric surfaces whose lattice automorphism group contains a non-trivial cyclic subgroup. We classify such surfaces and show that they all admit a cscK metric. We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modelled on a finite quotient of a toric affine variety with terminal singularities.

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov–Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet–Tevelev.

We show that the ring of regular functions of every smooth affine log Calabi–Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb {C}$-algebra structure with structure coefficients given by the geometric construction of Keel and Yu [The Frobenius structure theorem for affine log Calabi–Yau varieties containing a torus, Ann. Math. 198 (2023), 419–536]. To prove this result, we first give a canonical compactification of the mirror family associated with a pair $(Y,D)$ constructed by Gross, Hacking and Keel [Mirror symmetry for log Calabi–Yau surfaces I, Publ. Math. Inst. Hautes Ètudes Sci. 122 (2015), 65168] where $Y$ is a smooth projective rational surface, $D$ is an anti-canonical cycle of rational curves, and $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities. Then, we compute periods for the compactified family using techniques from Ruddat and Siebert [Period integrals from wall structures via tropical cycles, canonical oordinates in mirror symmetry and analyticity of toric degenerations, Publ. Math. Inst. Hautes Ètudes Sci. 132 (2020), 1–82] and use this to give a modular interpretation of the compactified mirror family.

Let S be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus g. We prove a quadratic upper bound on the genus g (i.e., $g\leq h\big (\chi (\mathcal {O}_S)\big )$, where h is a quadratic function). We also construct examples showing that the quadratic upper bounds cannot be improved to linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

We determine the cones of effective and nef divisors on the toroidal compactification of the ball quotient model of the moduli space of complex cubic surfaces with a chosen line. From this we also compute the corresponding cones for the moduli space of unmarked cubic surfaces.

In this article, we discuss the topology of varieties over $\mathbb {C}$, viz., their homology and homotopy groups. We show that the fundamental group of a quasi-projective variety has negative deficiency under a certain hypothesis on its second homology and therefore a large class of groups cannot arise as fundamental groups of varieties. For a smooth projective surface admitting a fibration over a curve, we give a detailed analysis of the homology and homotopy groups of their universal cover via a case-by-case analysis, depending on the nature of the singular fibers. For smooth, projective surfaces whose universal cover is holomorphically convex (conjecturally always true), we show that the second and third homotopy groups are free abelian, often of infinite rank.

Works by O’Grady allow to associate with a two-dimensional Gushel–Mukai (GM) variety, which is a K3 surface, a double Eisenbud–Popescu–Walter (EPW) sextic. We characterize the $K3$ surfaces whose associated double EPW sextic is smooth. As a consequence, we are able to produce symplectic actions on some families of smooth double EPW sextics which are hyper-Kähler manifolds.

We also provide bounds for the automorphism group of GM varieties in dimension 2 and higher.

KSB stability holds at codimension $1$ points trivially, and it is quite well understood at codimension $2$ points because we have a complete classification of $2$-dimensional slc singularities. We show that it is automatic in codimension $3$.

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$, these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$, we prove that they are different by exhibiting some walls.

Let $(S,L)$ be a general polarised Enriques surface, with L not numerically 2-divisible. We prove the existence of regular components of all Severi varieties of irreducible nodal curves in the linear system $|L|$, that is, for any number of nodes $\delta =0, \ldots , p_a(L)-1$. This solves a classical open problem and gives a positive answer to a recent conjecture of Pandharipande–Schmitt, under the additional condition of non-2-divisibility.

We introduce the notion of a simple fibration in $(1,2)$-surfaces – that is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$. We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds ‘on the Noether line’ with $K^3=\frac 43 p_g-\frac {10}3$, and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature.

The Gorenstein simple fibrations over ${\mathbb {P}}^1$ are Cartier divisors in a toric $4$-fold. This allows to us to show, among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space and that there is a second component when the geometric genus is congruent to $6$ modulo $8$; the threefolds in this component are new.

Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera $\overline P_1(V)$, $\overline P_2(V)$ and $\overline P_3(V)$ are equal to 1 and the logarithmic irregularity $\overline q(V)$ is equal to $2$. We prove that the quasi-Albanese morphism $a_V\colon V\to A(V)$ is birational and there exists a finite set S such that $a_V$ is proper over $A(V)\setminus S$, thus giving a sharp effective version of a classical result of Iitaka [12].

Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension $n \ge 2$ to a curve B defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi _f$. It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi _f> 0$, we prove that the general fibre F of f has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$. We also prove some sharper results of the same type under extra assumptions.

We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\mathrm {K}$-semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the Harder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138). One source of inspiration for our abstract study of Harder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984, Commentarii Mathematici Helvetic 59, 600–634). Another is the work of Faltings and Wüstholz (1994, Inventiones Mathematicae 116, 109–138), and Evertse and Ferretti (2013, Annals of Mathematics 177, 513–590), which is within the context of Diophantine approximation for projective varieties.

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 $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$, 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.

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi–Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim (Y)\leq 5$ and $Y$ is not of product type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of Kawamata log terminal pairs $(X, \Delta )$ with $K_X+\Delta$ numerically trivial and not of product type, in dimension at most four.

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along a specific $(4,\,4)$ curve. We show that, up to a finite group action, this stable pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.