Title: Arithmetic of the universal hypersurface
Abstract: The universal hypersurface X is the one defined by a polynomial with independent transcendental coefficients. A standard calculation shows that X can only have points of degrees divisible by deg X. In the case of curves one can easily say more: every degree kd point on the universal plane curve of degree d comes from a transversal intersection with a degree k curve. This fails, in general, in higher dimensions. I will report on work in progress, joint with Huang and Martin, which shows that a similar statement holds for degree kd points on a universal degree d>>k hypersurface. We build on recent techniques for analyzing degrees of irrationality of very general hypersurfaces, and combine them with a new theorem on the Cayley-Bacharach condition in the projective space.
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