Related papers: Galois embedding of K3 surface --abelian case--

For K\"ahler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal…

We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above…

It is shown that, the quasi-Koszulities of algebras and modules are Morita invariance. A finite-dimensional $K$-algebra $A$ with an action of $G$ is quasi-Koszul if and only if so is the skew group algebra $A \ast G$, where $G$ is a finite…

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for…

We describe the equations and Gr\"obner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The…

In this article, we study subloci of solvable curves in $\mathcal{M}_g$ which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus…

The Neron--Ogg--Safarevic criterion for abelian varieties tells that whether an abelian variety has good reduction or not can be determined from the Galois action on its l-adic etale cohomology. We prove an analogue of this criterion for…

We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.

We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface…

A bidouble cover is a flat $G:=\left(\mathbb{Z}/2\mathbb{Z}\right)^2$-Galois cover $X \rightarrow Y$. In this situation there exist three intermediate quotients $Y_1,Y_2$ and $Y_3$ which correspond to the three subgroups…

For an ample line bundle on an Abelian or K3 surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms…

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…

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…

In this note, we report some progress we made recently on the automorphisms groups of K3 surfaces. A short and straightforward proof of the impossibility of Z/(60) acting purely non-symplectically on a K3 surface, is also given, by using…

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse…

Motivated by the study of the growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one generalization, we give a result regarding the upper bound on the…

We give some positive answers to the following problem: Given a field $K$ and a continuous Galois representation $\rho:G_K \to GL_n(\mathbf{Q})$, construct an abelian variety $J/K$ of small dimension such that $\rho$ is a sub-representation…

We show the existence of abelian surfaces $A$ over $\mathbb{Q}_p$ having good reduction with supersingular special fibre whose associated $p$-adic Galois module $V_p(A)$ is not semisimple.

We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc…

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…