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What makes the Kronecker quiver special?

During the British Algebraic Geometry meeting last year in Warwick, Julian Holstein asked me an interesting question*:

What makes the Kronecker quiver special?

In the context of our conversation this meant the following: why is the derived category of the Kronecker quiver (with 2 arrows) equivalent to the derived category of the projective line, yet for the generalised Kronecker quiver (with n arrows, for n\neq 2) there is no equivalence with the derived category of any smooth projective variety?

The exact interpretation of this question is that the path algebra of the Kronecker quiver is the endomorphism algebra of some full and strong exceptional collection of objects in the derived category of a smooth projective variety, whilst the generalised Kronecker can never occur in such a fashion. So why is this the case?

There are many ways in which it is not hard to see that this is indeed the only possibility (arguing on algebraic K-theory, using Okawa’s indecomposability result for curves, studying exceptional objects, representation-theoretic arguments…) and certainly proving that the Kronecker quiver is indeed special wasn’t the issue. But somehow we felt that each of these arguments was like appealing to the classification of finite simple groups to prove some easy lemma in group theory. Yes, it works, but it isn’t the best method of proof.

Luckily, we are not just given a triangulated category: Serre duality imposes extra rigidity on a category, and it turns out that by studying the properties of the Serre functor we can distinguish between the Kronecker quiver and the generalised Kronecker quivers.

So for the actual answer: Bondal and Polishchuk (attributing the result to Suslin) prove in their paper Homological properties of associative algebras: the method of helices that the Serre functor on the level of the Grothendieck group (when equipped with the correct sign) is a unipotent operator. As the Gram matrix of the n-Kronecker quiver is given by

\displaystyle\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}

the Serre functor \kappa is given by

\displaystyle\begin{pmatrix} 1-n^2 & -n \\ n & 1 \end{pmatrix}

and the characteristic polynomial of -\kappa is given by t^2+(-n^2+2)t+1. For the matrix -\kappa to be unipotent we need the characteristic polynomial to be a power of t-1, hence n=2 is the only solution.

What I like about this answer is that it opens up questions in higher dimensions, to which I might return at some point.

* I don’t recall his exact words, but I guess we were using a more dg categorical lingo at the time, hence more likely they were something along:

What makes the gluing of two objects along a 2-dimensional vectorspace so special?

which gives a less entertaining title.

Cohomology tables in Macaulay2

Last year I created a little online tool to compute sheaf cohomology of twists of the structure sheaf on a complete intersection. It was a fun exercise in implementing power series computations and a result from SGA7 in JavaScript, but it unfortunately only applies to:

  • complete intersections
  • twists of the structure sheaf

Lately I have been toying a little with Macaulay2, and computations of this sort are very easy in this language, and can be done in far greater generality for arbitrary projective varieties and arbitrary coherent sheaves. As long as you can define the objects you are good to go, because of cohomologyTable. As you can use Macaulay2 online you might as well do it this way, instead of using the limited functionality in my implementation.

As an example you can run the following code (the online interface requires you to put your cursor on the first line and then repeatedly click evaluate):

loadPackage "BGG";

-- P^1
X = Proj(QQ[a,b])
cohomologyTable(OO_X(0) ++ OO_X(1), -4, 4)
cohomologyTable(cotangentSheaf X, -4, 4)
-- twisted cubic
X = Proj(QQ[a,b,c,d] / (a*c-b^2, b*d-c^2, a*d-b*c));
cohomologyTable(OO_X^1, -4, 4)
-- Fermat quartic (a K3 surface)
X = Proj(QQ[a,b,c,d] / (a^4+b^4+c^4+d^4));
cohomologyTable(tangentSheaf X, -4, 4)

The output for the K3 surface is

ii16 : -- Fermat quartic (a K3 surface)
       X = Proj(QQ[a,b,c,d] / (a^4+b^4+c^4+d^4));

ii17 : cohomologyTable(tangentSheaf X, -4, 4)

           -4 -3 -2 -1  0  1  2  3  4  5   6   7   8   9  10
oo17 = 2: 124 80 45 20  6  .  .  .  .  .   .   .   .   .   .
       1:   .  1  4 10 16 20 16 10  4  1   .   .   .   .   .
       0:   .  .  .  .  .  .  6 20 45 80 124 176 236 304 380

Observe the (weird) indexing: for the K3 surface we have \mathrm{h}^1(X,\mathrm{T}_X)=20, which sits in position (1,1) and not (1,0) as I would expect. The reason for this can be found in the manual:

This function takes as input a coherent sheaf F, two integers l and h, and prints the dimension dim HH^j F(i-j) for h>=i>=l.

A map of Calabi–Yau 3-folds

This morning I discovered a super awesome visualisation of Calabi–Yau 3-folds. The very same plot of Hodge numbers of Calabi–Yau 3-folds indicated 25 years ago that something like mirror symmetry could be true, and this tool is a spectacular visualisation of what is already known about mirror pairs for Calabi–Yau 3-folds. Make sure you read the documentation properly, and check out the “Open as static max zoom view” button, which gives you a plot of the whole view!

Remote access for MathSciNet

Have you ever encountered the username and password dialog box for MathSciNet whilst accessing it from your home connection? Of course, you could VPN to your institution to gain access, but it is now possible to sidestep this! I was told about this new functionality by Edward Dunne at the AMS Summer Institute in Algebraic Geometry, that allows you to use client-side storage to authenticate you for 90 days (it can be renewed after that), even if you are not using your institution’s connection! The relevant pages are:

  1. official explanation
  2. blog post announcement by Edward Dunne

Spread the word!

le superficie algebriche

An important result in algebraic geometry is the Kodaira–Enriques classification. It is a description of the geography of compact complex surfaces, later extended to surfaces in characteristic p. The Wikipedia page contains a wealth of information (seriously, it is one of the best mathematics pages in my opinion), and provides a pretty picture that describes the classification in an easy to remember way.

Together with Johan Commelin I made an interactive version of this picture, where you can get more information about the objects that are involved: more numerical invariants, a short description and references to the literature. You can find it at http://superficie.info. Johan has written a really detailed explanation on his weblog, you should check it out!

It is far from complete, in the sense that we certainly haven’t added every class of surfaces in the literature, and the existing descriptions can be improved. Neither of us is an expert in the birational classification of surfaces, but if you are and you want to contribute please get in touch with us! Also, if you want to contribute to the coding there is the GitHub project.


tl;dr: Go to superficie.info.

Disclaimer: I’m having issues with loading the page on my own computer, but Johan isn’t experiencing any problems. Please tell me if you experience any problems with loading. You should see a grid of dots with some lines on the left, with text and mathematics on the right.

New feature for the Stacks project: detailed tag edits

Because of the heat wave currently keeping Belgium in its grip (preventing me from doing research…) I decided to finally make the last fixes for a new feature for the Stacks project that has been in the make now for a while: detailed tag edits.

Git has a feature called git blame, that tells you when a certain line in a file was last edited and by whom. Starting from this idea Johan has built stacks-history that collects all this information throughout the lifetime of the Stacks project for all tags. If you add a little processing code to get this in the database in a meaningful format and some new website code, and you can check exactly when a tag has been changed!

If you want to see it in action, point your browsers to the history page for the tensor-Hom adjunction.

It’s not perfect (it might misinterpret some changes etc.) so if you spot mistakes, please do tell. And if you have suggestions to make this feature (or the Stacks project in general) better, please do tell!

Works of Sasha Rosenberg

The late Alexander (Sasha) Rosenberg, has a webpage collecting (some of) his work. Of course, most of his works were available through the preprint server of the Max–Planck Institute, but now things are easy accessible. It only went online a few days or weeks ago I think, and I guess it is maintained by his son Leo. There is also a book containing selected papers, which seems to be the print version of the website I linked.

Enjoy!

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