Colloquium: Tuesday May 16, 2023. Speaker: Assaf Hasson (BGU). Title: “How hard is it for a variety to forget who it is?”.

Our next Math colloquium this term will be in person on Tuesday the 16th of May, in room 614, Science & Education building.

A zoom link for our meetings is:

Speaker : Assaf Hasson (BGU)

Date : Tuesday, 16th of May, 2023.

Time : 14:00

Title: How hard is it for a variety to forget who it is?

Abstract: Consider an algebraic variety V over an algebraically closed field K (or, more generally, a finite Boolean combination of affine varieties over K). Consider V with its full Zariski structure. I.e., for all n we equip V^n with all the Zariski closed subsets of V^n defined over K. With a little work, it follows from a celebrated theorem of Hrushovksi and Zilber that in this structure a copy, K’,  of K is definable, that V can be identified with an algebraic variety over K’, and that the Zariski structure of Z over K’ is the same as the one over K we started with. We address the following question: suppose we start with V as above, but equip it with only part of its Zariski structure (e.g., if V is a curve, retain only one 3-dimensional Zariski closed subset of V^4). Under what conditions no information is actually lost, i.e., we can still recover every definable subset of V^n (all n) as a constructible set over some definable algebraically closed field K’.

In the talk I will present the notion of very ample strongly minimal sets, and explain how — using the recent solution of Zilber’s “restricted trichotomy conjecture” for algebraically closed fields — allows us to give a complete (abstract) answer the above question.

*Joint work with Ben Castle