No abstract

A conjecture of Beauville and Voisin states that for an irreducible symplectic variety X the subring of CH * (X ) generated by divisors goes injectively into the cohomology of X , via the cycle map. We prove this for a very general double Eisenbud-Popescu-Walter sextic.

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Let X be a Calabi-Yau threefold. We show that if there exists on X a non-zero nef non-ample divisor then X contains a rational curve, provided its second Betti number is greater than 4.

Let K be a number field. It is well known that the set of recurrencesequences with entries in K is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume {a n } is a recurrence sequence and suppose that all the a n have a d th root in the field K ; then (after possibly passing to a finite extension of K ) one can choose a sequence of such d th roots that satisfies a recurrence itself. This was proved true in a preceding paper of the second author. In this article we generalize this result to more general monic equations; the former case can be recovered for g(X, Y ) = X d − Y = 0. Combining this with the Hadamard quotient theorem by Pourchet and Van der Poorten, we are able to get rid of the monic restriction, and have a theorem that generalizes both results.