Lubjana Beshaj

Decomposition of Some Jacobian Varieties of Dimension 3

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over ℂ. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve E and degree 4 covers to elliptic curves E 1 and E 2 is a 2-dimensional subvariety of the hyperelliptic moduli \({\mathcal H}_3\). We determine this subvariety explicitly. For any given moduli point \(\wp \in{\mathcal H}_3\) we determine explicitly if the corresponding genus 3 curve \(\mathcal{X}\) belongs or not to such family. When it does, we can determine elliptic subcovers E, E 1, and E 2 in terms of the absolute invariants t 1, …, t 6 as in [12]. This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split. The sublocus of such family when E 1 ≅ E 2 is a 1-dimensional variety which we determine explicitly. We can also determine \(\mathcal{X}\) and E starting form the j-invariant of E 1.

On Prym varieties for the coverings of some singular plane curves

Let k be a field of characteristic zero containing a primitive nth root of unity. Let

n

C^0_n

Singular locus on the space of genus 2 curves with decomposable Jacobians

Cn0 be a singular plane curve of degree n over k admitting an order n automorphism, n nodes as the singularities, and

Equations for superelliptic curves over their minimal field of definition

C_n

Heights on algebraic curves.

Cn be its normalization. In this paper we study the factors of Prym variety