Olivier Martin, Measures of Association for Algebraic Varieties

I will discuss recent work in collaboration with R. Lazarsfeld which explores the following question: Given varieties X and Y of the same dimension how far are they from being birational? I will define various “measures of association” which quantify the failure of X and Y to be birational and present partial results, heuristics, and several open problems. For instance, given an n-fold Z dominating very general hypersurfaces X and Y in Pⁿ⁺¹ of degrees d,e > 2n+1, we show that the degrees of the projections Z—>X and Z—>Y are at least d and e. Moreover, given very general hyperelliptic curves X and Y, any hyperelliptic curve in XxY is contracted by the projection to X or the projection to Y.