Intersections and the Bezout Range

Let A be a simple abelian variety, and X and Y be two subvarieties. We say X and Y are in the Bezout range if \dim X + \dim Y >= \dim A, and outside of the Bezout range otherwise. It is known that two varieties in the Bezout range in A always intersect. In this talk we explain that, after multiplying Y by some endomorphism n, we can make them intersect properly, and moreover such intersections give an analytically dense set of intersections with X. Moreover in the case where X and Y are outside the Bezout range, we show that X and n Y almost never intersect, except in the presence of torsion points.

Joint work with Greg Baldi.