Zariski Geometries Summer Seminar

2005-06-02T10:04:00.000-07:00

Question: (On exercise 5.8 3) Is there any example of an irreducible set $F$ such that there's a power of $F$ which is NOT irreducible? For constructible sets, that's not possible.

2005-05-23T10:00:00.000-07:00

Today we discussed example 4.8 and solved the exercises 4.10 and 4.11. In the solution of 4.10 the main ingredient is quantifier elimination. This allowed us to imitate the proofs in algebraic geometry. In 4.11 we assumed $\omega$-saturation (which we can wlog) and worked with automorphisms.

More comments on section 4.

2005-05-18T21:57:00.000-07:00

Sonat noticed that a good definition of locus(a/A) depends of one of two things:

Either you have essential uncountability (EU) in $M$ (that implies $\omega_1$-compactness in $M$ and so DCC in any elementary extension) , or
$A$ is a subset of $M$ (in which case the locus is actually a basic predicate).

Finally we decided, from now on, to assume that when it is mentioned an $A$ set, $A\subseteq M$. So far, this assumption has solved most of our problems with the proofs.

An issue that came up today while we were discussing what we're gonna talk about tomorrow is the distinction between predicates and actual sets. Zilber uses both notations indistinctively and sometimes it is hard to dilucidate which one is the object he is talking about.

May 19th session.

2005-05-17T14:46:00.000-07:00

On May 19th. Sonat and I will be talking about section 4. We will go over some of the proof and, If time permits, we will even discuss with the attendands a few exercises.

A comment on Section 4

2005-05-17T13:15:00.000-07:00

Sonat has pointed out that the definition of dim(a/A) (Def. 4.3) leaves a bit to be desired, as gives the definition in terms of the dimension of the locus of a over A, which is not itself defined.

The issue is that the beginning of section 4 considers a situation where we do not have DCC for what Zilber calls "closed sets", and which I will hereafter refer to as "basic closed sets". He has defined dimension for basic closed sets, but not for intersections of such (which I will refer to as "closed sets").

But Def 4.3 can be easily repaired by defining the dimension of a closed set to be the minimum dimension of any basic closed set containing it.

Also note that in the proof of Lemma 4.6, when Zilber refers to the closure of a particular set, he has in mind the smallest closed set containing that set, rather than the smallest basic closed set (which will not, in general, exist).

First meeting and general info.

2005-05-10T13:49:00.000-07:00

Hello, this is a weblog to complement the summer seminar on Zariski Geometries we will be running during summer 2005.

The notes we will follow are available at http://www.maths.ox.ac.uk/~zilber/s.pdf.

The first meeting will be on friday the 13th at 4:00 pm at Crane Alley. An informative e-mail will be sent to the Urbana logic group announcing it.

I guess we can use this space to post questions we might come up with during the sessions and have some extra discussion. If you're interested on being able to post entries here, you should create a blogger account and then contact me so I can add you to the list of contributors of the blog. Of course, everybody, even if you're not one of the aformentioned contributors, can write comments clicking on the comment link at the end of each entry and filling up the corresponding form.

May the force be with us.