Related papers: Geometry and arithmetic of certain log K3 surfaces

We consider algebraic actions of a cyclic group of order p on a K3 surface defined over an algebraically closed field of characteristic p. We classify possible loci of fixed points as well as possible quotient surfaces.

We study the fixed singularities imposed on members of a linear system of surfaces in P^3_C by its base locus Z. For a 1-dimensional subscheme Z \subset P^3 with finitely many points p_i of embedding dimension three and d >> 0, we determine…

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo…

In this paper we give for all $n \geq 2$, d>0, $g \geq 0$ necessary and sufficient conditions for the existence of a pair (X,C), where X is a K3 surface of degree 2n in $\matbf{P}^{n+1}$ and C is a smooth (reduced and irreducible) curve of…

We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite…

Surfaces of general type with geometric genus $p_g=0$, which can be given as Galois covering of the projective plane branched over an arrangement of lines with Galois group $G=(\mathbb Z/q\mathbb Z)^k$, where $k\geq 2$ and $q$ is a prime…

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We…

For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces…

If $S$ is a quintic surface in $\mathbb P^3$ with singular set $15$ $3$-divisible ordinary cusps, then there is a Galois triple cover $\phi:X\to S$ branched only at the cusps such that $p_g(X)=4,$ $q(X)=0,$ $K_X^2=15$ and $\phi$ is the…

Let $X$ be a normal crossing compact complex surface with triple points. We prove that there exists a family of smoothings of $X$ when $X$ satisfies suitable conditions. Since our differential geometric proof also includes the case where…

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree~$2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth…

Let $X$ be a K3 or Enriques surface with good reduction. Let $G$ be a finite group acting (not necessarily linearly) on $X$. We give a criterion for this group action to extend to a smooth model of $X$ in terms of the action of $G$ on the…

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces…

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on…

Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how…

Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has non-trivial, cyclic Sylow…

We examine the finite group actions on K3 and Abelian surfaces giving the same orbit space after desingularization. We show that when the group is not Z_2, then the Picard number of the K3 surface must be 19 or 20, and that in the latter…

If $X'= ({\mathbb B} / \Gamma)'$ is a torsion free toroidal compactification of a discrete ball quotient $X_o={\mathbb B} / \Gamma$ and $\xi : (X', T = X'\setminus X_o) \rightarrow (X, D = \xi (T))$ is the blow-down of the $(-1)$-curves to…

We characterize all projective K3 surfaces on which every integral pseudoeffective divisor admits an integral Zariski decomposition, using an explicit, terminating finite-step algorithm.

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has…