Aug 13, 2020
Thursday

01:30 PM  03:30 PM


Fellowship of the Ring, National Seminar: CayleyBacharach theorems and measures of irrationality
Brooke Ullery (Emory University)

 Location
 MSRI: Online/Virtual
 Video

 Abstract
To attend this seminar, you must register in advance, by clicking HERE.
If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes Z linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original CayleyBacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the socalled CayleyBacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.
 Supplements



