We consider the question of when a Jacobian of a curve of genus $2g$ admits a $(2,2)$-isogeny to a product of two polarized dimension g abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is hyperelliptic, so is the other.

For $g=2,$ this allows us to describe $(2,2)$-decomposable genus $4$ Jacobians in terms of Prym varieties. We describe the locus of such genus $4$ curves in terms of the geometry of the Igusa quartic threefold. We also explain how our characterization relates to Prym varieties of unramified double covers of plane quartic curves, and we describe this Prym map in terms of $6$ and $7$ points in $\mathbb {P}^3$.

We also investigate which genus $4$ Jacobians admit a $2$-isogeny to the square of a genus $2$ Jacobian and give a full description of the hyperelliptic ones. While most of the families we find are of the expected dimension $1$, we also find a family of unexpectedly high dimension $2$.

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.

Let C be a family of curves over a non-singular variety S. We study algebraic cycles on the relative symmetric powers C[n] and on the relative Jacobian J. We consider the Chow homology CH*(C[∙]/S) := ⊕n CH*(C[n]/S) as a ring using the Pontryagin product. We prove that CH*(C[∙]/S) is isomorphic to CH*(J/S)[t]〈u〉, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over CH*(J/S). We give two such isomorphisms that over a general base are different. Further we give precise results on how CH*(J/S) sits embedded in CH*(C[∙]/S) and we give an explicit geometric description of how the operators and ∂u act. This builds upon the study of certain geometrically defined operators Pi,j (a) that was undertaken by one of us.

Our results give rise to a new grading on CH*(J/S). The associated descending filtration is stable under all operators [N]*, and [N]* acts on as multiplication by Nm. Hence, after − ⊗ ℚ this filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's.

Finally, we give a version of our main result for tautological classes, and we show how our methods give a simple geometric proof of some relations obtained by Herbaut and van der Geer–Kouvidakis, as later refined by one of us.

It is known that ℚ-derived univariate polynomials (polynomials defined over ℚ, with the property that they and all their derivatives have all their roots in ℚ) can be completely classified subject to two conjectures: that no quartic with four distinct roots is ℚ-derived, and that no quintic with a triple root and two other distinct roots is ℚ-derived. We prove the second of these conjectures.

AMS 2000 Mathematics subject classification: Primary 11G30

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.

For $x\in (a_{j-1}, a_j)\ (j=1,\ldots, p+1;\ a_0\!:=-\infty, \ a_{p+1} \!:=\infty)$ the mapping $T_j\!: w=x-\sum ^p_{l=1}\lambda _l/(x-a_l)\ (\lambda _l$>$0, \ a_l\in$R) is onto R. It was shown by G. Boole in the 1850's that $\sum ^{p+1} _{j=1}[(\partial w/\partial x) ^{-1}] _{x=T^{-1}_j(w)}=1.$ We give an n-dimensional analogue of this result. The proof makes use of the Griffiths–Harris residue theorem from algebraic geometry.

A strategy is proposed for applying Chabauty's Theorem to hyperelliptic curves of genus ${>} 1$. In the genus 2 case, it shown how recent developments on the formal group of the Jacobian can be used to give a flexible and computationally viable method for applying this strategy. The details are described for a general curve of genus 2, and are then applied to find ${\bm C}({\bb Q})$ for a selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.