Related papers: Genus two Veech surfaces arising from general quad…

We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli…

We study the geometry of quartic surfaces in IP^3 that contain a line of the second kind over algebraically closed fields of characteristic different from 2,3. In particular, we correct Segre's claims made for the complex case in 1943.

Two families of surfaces arise from considering cyclic branched covers of $\mathbb{P}^{2}$ over smooth quartic curves. These consist of degree 2 del Pezzo surfaces with a $\mathbb{Z}/2\mathbb{Z}$ action and $K3$ surfaces with a…

The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of base Riemann surfaces and fibers. We also show a relation between types of…

We study deformations of a genus one Riemann surface and of a second order Abelian differential on the surface which preserve the periods of the differential with respect to a chosen canonical homology basis of the surface. We call these…

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic…

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…

Using the Schwarzian derivative we construct a sequence $\left(P_{\ell}\right)_{\ell \geqslant 2}$ of meromorphic differentials on every non-flat oriented minimal surface in Euclidean $3$-space. The differentials…

We construct absolutely simple jacobians of non-hyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1. This paper is a natural continuation of author's paper math.AG/0405156.

In this note we present examples of complex algebraic surfaces with canonical maps of degree $12$, $13$, $15$, $16$ and $18$. They are constructed as quotients of a product of two curves of genus $10$ and $19$ using certain non-free actions…

In this paper we study principally polarized complex abelian varieties that admit an automorphism of order 3. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do…

In this paper, we study algebraic surfaces of general type with $p_g=q=1$ and genus 2 Albanese fibrations. We first study the examples of surfaces with $p_g=q=1, K^2=5$ and genus 2 Albanese fibrations constructed by Catanese using singular…

We discuss several geometric features of a Kummer surface associated with a (1,2)-polarized abelian surface defined over the field of complex numbers. In particular, we show that any such Kummer surface can be modeled as the double cover of…

Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$…

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine…

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate…

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very…

We show that if A is a d-dimensional abelian variety in a smooth quadric of dimension 2d then d=1 and A is an elliptic curve of bidegree (2,2) on a quadric. This extends a result of Van de Ven which says that A only can be embedded in…