We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are Kawamata log terminal (klt) if and only if is finitely generated. We introduce a notion of nefness for non-ℚ-Gorenstein varieties and study some of its properties. We then focus on these properties for non-ℚ-Gorenstein toric varieties.

We show that the pair (X, –KX ) is K-unstable for a del Pezzo manifold X of degree 5 with dimension 4 or 5. This disproves a conjecture of Odaka and Okada.

Let X be a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles on X obtained by an augmentation from a strong exceptional collection on the minimal model of X. In particular, we calculate canonical DG algebras of smooth toric surfaces.

Let $S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree $d\geq 3$ . In this note, we classify stable Ulrich bundles on $S$ of rank two. We also study their moduli spaces.

To a pair of elliptic curves, one can naturally attach two K3 surfaces: the Kummer surface of their product and a double cover of it, called the Inose surface. They have prominently featured in many interesting constructions in algebraic geometry and number theory. There are several more associated elliptic K3 surfaces, obtained through base change of the Inose surface; these have been previously studied by Masato Kuwata. We give an explicit description of the geometric Mordell–Weil groups of each of these elliptic surfaces in the generic case (when the elliptic curves are non-isogenous). In the nongeneric case, we describe a method to calculate explicitly a finite index subgroup of the Mordell–Weil group, which may be saturated to give the full group. Our methods rely on several interesting group actions, the use of rational elliptic surfaces, as well as connections to the geometry of low degree curves on cubic and quartic surfaces. We apply our techniques to compute the full Mordell–Weil group in several examples of arithmetic interest, arising from isogenous elliptic curves with complex multiplication, for which these K3 surfaces are singular.

Let $(X,\unicode[STIX]{x1D6E5})$ be an $n$ -dimensional $\unicode[STIX]{x1D716}$ -klt log $\mathbb{Q}$ -Fano pair. We give an upper bound for the volume $\text{Vol}(X,\unicode[STIX]{x1D6E5})=(-(K_{X}+\unicode[STIX]{x1D6E5}))^{n}$ when $n=2$ , or $n=3$ and $X$ is $\mathbb{Q}$ -factorial of $\unicode[STIX]{x1D70C}(X)=1$ . This bound is essentially sharp for $n=2$ . The main idea is to analyze the covering families of tigers constructed in J. McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov–Alexeev–Borisov Conjecture, which asserts boundedness of the set of $\unicode[STIX]{x1D716}$ -klt log $\mathbb{Q}$ -Fano varieties of a given dimension $n$ .

In this paper, we exhibit explicit automorphisms of maximal Salem degree 22 on the supersingular K3 surface of Artin invariant one for all primes $p\equiv 3~\text{mod}\,4$ in a systematic way. Automorphisms of Salem degree 22 do not lift to any characteristic zero model.

For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We define an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. Then we study rational maps of moduli spaces of positive height to moduli spaces of representation of quivers, effective cones of moduli spaces, and strange duality for height-zero moduli spaces.

We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are $p$ -adic in nature.

We compute the complete set of candidates for the zeta function of a K $3$ surface over $\mathbb{F}_{2}$ consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over $\mathbb{F}_{2}$ . These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K $3$ surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

Let $S$ be a smooth minimal complex surface of general type with $p_{g}=0$ and $K^{2}=7$ . We prove that any involution on $S$ is in the center of the automorphism group of $S$ . As an application, we show that the automorphism group of an Inoue surface with $K^{2}=7$ is isomorphic to $\mathbb{Z}_{2}^{2}$ or $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ . We construct a $2$ -dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$ .

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

We prove that for any prime number $p\geqslant 3$ , there exists a positive number $\unicode[STIX]{x1D705}_{p}$ such that $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})\geqslant \unicode[STIX]{x1D705}_{p}c_{1}^{2}$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$ . In particular, $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})>0$ . This answers a question of Shepherd-Barron when $p\geqslant 3$ .

An invertible polynomial in n variables is a quasi-homogeneous polynomial consisting of n monomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau–Ginzburg models, Berglund, Hübsch and Henningson considered a pair (f, G) consisting of an invertible polynomial f and an abelian group G of its symmetries together with a dual pair . Here we study the reduced orbifold zeta functions of dual pairs (f, G) and and show that they either coincide or are inverse to each other depending on the number n of variables.

In this paper a new geometric characterization of the $n$ th symmetric product of a curve is given. Specifically, we assume that there exists a chain of smooth subvarieties $V_{i}$ of dimension $i$ , such that $V_{i}$ is an ample divisor in $V_{i+1}$ and its intersection product with $V_{1}$ is one; that the Albanese dimension of $V_{2}$ is $2$ and the genus of $V_{1}$ is equal to the irregularity of the variety. We prove that in this case the variety is isomorphic to the symmetric product of a curve.

In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_{S})$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_{S}$ and such that the open set $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$-actions on their affine cones defined by their anticanonical divisors.

An affine symplectic singularity $X$ with a good $\mathbf{C}^{\ast }$-action is called a conical symplectic variety. In this paper we prove the following theorem. For fixed positive integers $N$ and $d$, there are only a finite number of conical symplectic varieties of dimension $2d$ with maximal weights $N$, up to an isomorphism. To prove the main theorem, we first relate a conical symplectic variety with a log Fano Kawamata log terminal (klt) pair, which has a contact structure. By the boundedness result for log Fano klt pairs with fixed Cartier index, we prove that conical symplectic varieties of a fixed dimension and with a fixed maximal weight form a bounded family. Next we prove the rigidity of conical symplectic varieties by using Poisson deformations.

We develop a new non-vanishing theorem for del Pezzo surfaces with quotient singularities.

We study cup products in the integral cohomology of the Hilbert scheme of $n$ points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.

Supplementary materials are available with this article.