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We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang and To on projective normality.
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose is an elliptic curve over a field , and are points on defined over . To this information we associate an -dimensional array of values in satisfying a nonlinear recurrence relation. Arrays satisfying this relation are called elliptic nets. We demonstrate an explicit bijection between the set of elliptic nets and the set of elliptic curves with specified points. We also obtain Laurentness/integrality results for elliptic nets.
We give a concrete description of the category of étale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical -typical and big Witt vector functors but also for certain analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for classical Witt vectors. Our larger purpose is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces, so the basics are developed in some detail, with an eye toward future applications.