John Baez, Two Miracles of Algebraic Geometry is a really nice post (what else did you expect from John!) by John Baez where he explains the categorical properties of the Albanese variety as a functor. It really highlights why algebraic geometry is special, and it does so by asking this seemingly naive MathOverflow question, in which he does something that was never envisioned by the Italian school of algebraic geometry, namely turning the Albanese variety into an endofunctor.
A funny quote from the blogpost is
But forgetful functors often go unspoken in ordinary mathematical English: they’re not just forgetful, they’re forgotten.
Shane Kelly, Some observations about motivic tensor triangulated geometry over a finite field is a set of notes for a summer school from last year, which was my first real introduction to the stable homotopy category and why an algebraic geometer could be interesting in all this. It applies all the (proven) machinery to reduce very abstract questions about the tensor triangulated geometry of the stable homotopy category to the (for me at least) little less abstract category of Voevodsky motives, and then appeals to certain conjectures in the special case of finite fields to compute that the Balmer spectrum of this topological gadget is actually !
At the moment I am working on something which involves properties of non-reduced curves, and I realised that there are (at least) two interpretations you can give to the term fuzzy projective line. This confused me at first, but it’s all very simple actually:
- You can consider .
- You can also consider .
These are not the same. One way of seeing this is by computing the global sections of the structure sheaf. For the first you can for instance appeal to Hartshorne, exercise III.5.5, because this fuzzy projective line is a complete intersection, and we get . On the other hand we have by construction that .
What is happening is that the second fuzzy projective line is not a complete intersection, hence we cannot embed it into . But it is not hard to realise it inside by using that is a closed subscheme of , so is a closed subscheme of , which we can embed into . An explicit set of equations cutting out inside would be .
At this year’s ICRA in Syracuse the ICRA award was given to Greg Stevenson and Qiu Yu. Congratulations!
I ran into the following lovely quote by the late Andrei Tyurin:
An algebraic geometer is skillful enough if he or she can recognize the geometric person under many guises of different dimensions.
The reason for this quote (and the quote itself) can be read in his 1995 lecture notes for the summer school on algebraic geometry in Ankara. One of the examples that a skilful algebraic geometer must know are the objects:
- in dimension 0: 6 distinct points on the projective line up to projective equivalence
- in dimension 1: a genus 2 curve
- in dimension 2: a cubic surface with a unique ordinary double point
- in dimension 3: a nonsingular intersection of 2 quadric hypersurfaces in
and more importantly their relation to each other:
- the genus 2 curve is a double cover of the projective line ramified in 6 distinct points
- blowing up 6 points on a conic (so the points are not in general position!) in gives a singular cubic surface
- the pencil spanned by the two quadric hypersurfaces has 6 singular members
Recently, many developments in the study of derived categories of smooth projective varieties have been of this nature too, e.g. Kuznetsov’s homological projective duality, or the Segal–Thomas example of a Calabi–Yau threefold embedding in a Fano elevenfold. In noncommutative algebraic geometry there is also the intricate connection between noncommutative planes and planar cubic curves which comes to mind.
He uses the word person at another point in the text where I would say object. I don’t know whether he has his own idiosyncratic vocabulary just like Erdős did.
Cox, Toric varieties are the lectures notes that David Cox wrote for his lecture series at this year’s GAeL. I wasn’t there for the conference, but the lectures notes are definitely worth a read.
Woit, Quantum Theory and Representation Theory, the Book is the blog post announcing Peter’s upcoming book. The pdf of the current version is freely available. This is one of the things I would like to learn more about, if only I could read and understand things faster.
To end this fortnight’s list of links: The Canadian who reinvented mathematics is a nice and interesting piece about Robert Langlands. It’s a bit annoying though how the journalist likes to stress how little he remembers from high school mathematics…