Related papers: Isogenous components of Jacobian surfaces

Let $X_1, ..., X_m$ denote smooth projective curves of genus $g_i \geq 2$ over an algebraically closed field of characteristic 0 and let $n$ denote any integer at least equal to $1+\max_{i=1}^m g_i$. We show that the product $JX_1 \times…

Let $\mathbb{K}$ be an algebraically closed field of characteristic $p>0$, and let $C$ be a nonsingular projective curve over $\mathbb{K}$. We prove that for any real number $x \geq 2$, there are minimal surfaces of general type $X$ over…

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of…

Given a symplectic three-fold $(M,\omega)$ we show that for a generic almost complex structure $J$ which is compatible with $\omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class…

We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds)…

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from…

A threefold isogenous to a product of curves $X$ is a quotient of a product of three compact Riemann surfaces of genus at least two by the free action of a finite group. In this paper we study these threefolds under the assumption that the…

A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete…

Let p:C-->Y be a covering of smooth, projective curves which is a composition of \pi:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the…

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one…

This paper deals with singularities of genus 2 curves on a general (d_1,d_2)-polarized abelian surface (S,L). In analogy with Chen's results concerning rational curves on K3 surfaces [Ch1,Ch2], it is natural to ask whether all such curves…

Suppose that $K$ is a field of characteristic 0, $K_a$ is its algebraic closure, $p$ is a prime, $q=p^r$ is a power prime. Suppose that $f(x) \in K[x]$ is a polynomial of degree $n > 4$ without multiple roots. Let us consider the…

Given a genus $2$ curve $C$ with a rational Weierstrass point defined over a number field, we construct a family of genus $5$ curves that realize descent by maximal unramified abelian two-covers of $C$, and describe explicit models of the…

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…

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no…

Let $X$, $D$ be a smooth projective surface and a simple normal crossing divisor on $X$, respectively. Suppose $\kappa (X, K_X + D)\ge 0$, let $C$ be an irreducible curve on $X$ whose support is not contained in $D$ and $\alpha$ a rational…

The nonabelian Jacobian $\JA$ of a smooth projective surface $X$ is inspired by the classical theory of Jacobian of curves. It is built as a natural scheme interpolating between the Hilbert scheme $\XD$ of subschemes of length $d$ of $X$…

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is…

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles…

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves.…