Related papers: Geometry and arithmetic of certain log K3 surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a d-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for p any…

If G is a complex semisimple algebraic group, we characterize the normality and the smoothness of its simple linear compactifications, namely those equivariant GxG-compactifications which possess a unique closed orbit and which arise in a…

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism…

The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in…

Consider a rational elliptic surface over a field $k$ with characteristic $0$ given by $\mathcal{E}: y^2 = x^3 + f(t)x + g(t)$, with $f,g\in k[t]$, $\text{deg}(f) \leq 4$ and $\text{deg}(g) \leq 6$. If all the bad fibres are irreducible,…

Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the N\'eron-Severi…

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo…

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition…

We study the action of the Klein simple group G consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona…

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, \Delta)$ satisfying the following properties: (i) $X$ is a projective surface defined over an…

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's…

We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute $G$-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups…

The K-moduli theory provides a different compactification of moduli spaces of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces…

For every supersingular $K3$ surface $X$ in characteristic 2, there exists a homogeneous polynomial $G$ of degree 6 such that $X$ is birational to the purely inseparable double cover of a projective plane defined by $w^2=G$. We present an…

For an arithmetic surface X and a Weil divisor $D$, there are natural arithmetic cohomology groups $H_{\mathrm{ar}}^i(X, \mathcal O_X (D))$ $(i=0,1,2)$. Using ind-pro topology on adelic space $\mathbb A_{X, 012}^{\mathrm{ar}}$, we show that…

We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the…

We study the moduli space $\mathcal{F}_{T_1}$ of quasi-trielliptic K3 surfaces of type I, whose general member is a smooth bidegree $(2,3)$-hypersurface of $\mathbb{P}^1\times \mathbb{P}^2$. Such moduli space plays an important role in the…

Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the…

A variety X with an action of a finite group G is said to be G-unirational if there is a G-equivariant dominant rational map V -> X where V is a faithful linear representation of G. This generalizes the usual notion of unirationality. We…

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$ action. In particular, we describe families whose general member contains $8,16,24$ or $32$…