Dedicated to Roy Smith on his 65th birthday. Abstract. We study complex abelian varieties of dimension g that have a vector bundle of rank g with a section that vanishes at a single point, with multiplicity 1. The authors of [PSP] study which varieties X (defined, say, over an algebraically closed field) satisfy what they call "the diagonal property": there exists a vector bundle of rank dim(X) on X × X with a section whose zero-scheme is the diagonal. This property is known to hold for all

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... varieties SL n /P ([F1]) and, using a variant of Serre's construction, they show that it holds for projective surfaces which have a cohomologically trivial line bundle. In particular, it holds for abelian surfaces. The diagonal property implies the weaker "point property": for each point x of X, there exists a vector bundle of rank dim(X) on X with a section whose zeroscheme is x. When X is a group variety, these two properties are equivalent (it is even enough to have the point property for a single point x). In this note, we study these (equivalent) properties when X is a complex abelian variety. We show in §1 that a general non-principally polarized abelian variety of a sufficiently high dimension does not have these properties. Using Picard bundles, we show in §2 that these properties hold when X is the Jacobian of a smooth curve (or a product of such). However, Lange pointed out that in any dimension, principally polarized abelian varieties that have the point property are dense in their moduli space (Remark 2.4). I do not know any principally polarized abelian variety that does not have the point property, although I would like to think that Jacobians of curves are the only principally polarized abelian varieties with Picard number 1 that have this property, thereby giving us another (partial) solution to the Schottky problem. In §3 and 4, we prove a necessary condition for the point property to hold, and use it to get restrictions on possible vector bundles. We work over the complex numbers, although the results of §2 and 3 are valid over an algebraically closed field of arbitrary characteristic.