# Olivier Martin: Measures of association for algebraic varieties

**Time: **
Thu 2021-12-09 14.00 - 15.00

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan (alt. Zoom, meeting ID: 921 756 1880)

**Participating: **
Olivier Martin (Stony Brook University)

**Abstract:** 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 \(\mathbb{P}^{n+1}\) of degrees \(d,e>2n+1\), we show that the degrees of the projections \(Z\to X\) and \(Z\to Y\) are at least *d* and *e*. Moreover, given very general hyperelliptic curves *X* and *Y*, any hyperelliptic curve in \(X\times Y\) is contracted by the projection to *X* or the projection to *Y*.