The Vector Decomposition Problem for Elliptic and Hyperelliptic Curves

The group of m-torsion points on an elliptic curve, for a prime number m, forms a two-dimensional vector space. It was suggested and proven by Yoshida that under certain conditions the vector decomposition problem (VDP) on a two-dimensional vector space is at least as hard as the computational Diffie-Hellman problem (CDHP) on a one-dimensional subspace. In this work we show that even though this assessment is true, it applies to the VDP for m-torsion points on an elliptic curve only if the curve is supersingular. But in that case the CDHP on the one-dimensional subspace has a known sub-exponential solution. Furthermore, we present a family of hyperelliptic curves of genus two that are suitable for the VDP

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces $S$ of nonnegative Kodaira dimension all having constant geometric genus $g \geq 2$ and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is ``vector bundle-free'' and relies on deformation theory and bending-and-breaking of rational curves in \Sym^2(S). We also give examples of families of such curves.Comment: 18 page

On the Chow ring of Cynk-Hulek Calabi-Yau varieties and Schreieder varieties

This note is about certain locally complete families of Calabi-Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow-Kuenneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology, via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk-Hulek Calabi-Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk-Hulek Calabi-Yau varieties provide examples of Calabi-Yau varieties with "degenerate" motive.Comment: to appear in the Canadian Journal of Mathematic

The Hodge Conjecture for general Prym varieties

The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.Comment: LaTeX-fil

Covering of elliptic curves and the kernel of the Prym map

Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map $\Phi$. More precisely, we describe the codifferential of the period map $P$ associated to $\Phi$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the coverings for which the dimension of $\ker(dP)$ is the least possibile. This is useful in order to exclude the existence of non isotrivial fibrations with maximal relative irregularity and thus also in order to give counterexamples to the Xiao's conjecture mentioned above. The first counterexample to the original conjecture, due to Pirola, is then analysed in our framework.Comment: 21 pages, no figures. The seminal ideas at the base of this article were born in the framework of the PRAGMATIC project of year 201