Quiver Grassmannians can be anything

A standard Grassmannian $Gr(m,V)$ is the manifold having as its points all possible $m$-dimensional subspaces of a given vectorspace $V$. As an example, $Gr(1,V)$ is the set of lines through the origin in $V$ and therefore is the projective space $\mathbb{P}(V)$. Grassmannians are among the nicest projective varieties, they are smooth and allow a cell decomposition.

A quiver $Q$ is just an oriented graph. Here’s an example

A representation $V$ of a quiver assigns a vector-space to each vertex and a linear map between these vertex-spaces to every arrow. As an example, a representation $V$ of the quiver $Q$ consists of a triple of vector-spaces $(V_1,V_2,V_3)$ together with linear maps $f_a~:~V_2 \rightarrow V_1$ and $f_b,f_c~:~V_2 \rightarrow V_3$.

A sub-representation $W \subset V$ consists of subspaces of the vertex-spaces of $V$ and linear maps between them compatible with the maps of $V$. The dimension-vector of $W$ is the vector with components the dimensions of the vertex-spaces of $W$.

This means in the example that we require $f_a(W_2) \subset W_1$ and $f_b(W_2)$ and $f_c(W_2)$ to be subspaces of $W_3$. If the dimension of $W_i$ is $m_i$ then $m=(m_1,m_2,m_3)$ is the dimension vector of $W$.

The quiver-analogon of the Grassmannian $Gr(m,V)$ is the Quiver Grassmannian $QGr(m,V)$ where $V$ is a quiver-representation and $QGr(m,V)$ is the collection of all possible sub-representations $W \subset V$ with fixed dimension-vector $m$. One might expect these quiver Grassmannians to be rather nice projective varieties.

However, last week Markus Reineke posted a 2-page note on the arXiv proving that every projective variety is a quiver Grassmannian.

Let’s illustrate the argument by finding a quiver Grassmannian $QGr(m,V)$ isomorphic to the elliptic curve in $\mathbb{P}^2$ with homogeneous equation $Y^2Z=X^3+Z^3$.

Consider the Veronese embedding $\mathbb{P}^2 \hookrightarrow \mathbb{P}^9$ obtained by sending a point $(x:y:z)$ to the point

\[ (x^3:x^2y:x^2z:xy^2:xyz:xz^2:y^3:y^2z:yz^2:z^3) \]

The upshot being that the elliptic curve is now realized as the intersection of the image of $\mathbb{P}^2$ with the hyper-plane $\mathbb{V}(X_0-X_7+X_9)$ in the standard projective coordinates $(x_0:x_1:\cdots:x_9)$ for $\mathbb{P}^9$.

To describe the equations of the image of $\mathbb{P}^2$ in $\mathbb{P}^9$ consider the $6 \times 3$ matrix with the rows corresponding to $(x^2,xy,xz,y^2,yz,z^2)$ and the columns to $(x,y,z)$ and the entries being the multiplications, that is

$$\begin{bmatrix} x^3 & x^2y & x^2z \\ x^2y & xy^2 & xyz \\ x^2z & xyz & xz^2 \\ xy^2 & y^3 & y^2z \\ xyz & y^2z & yz^2 \\ xz^2 & yz^2 & z^3 \end{bmatrix} = \begin{bmatrix} x_0 & x_1 & x_2 \\ x_1 & x_3 & x_4 \\ x_2 & x_4 & x_5 \\ x_3 & x_6 & x_7 \\ x_4 & x_7 & x_8 \\ x_5 & x_8 & x_9 \end{bmatrix}$$

But then, a point $(x_0:x_1: \cdots : x_9)$ belongs to the image of $\mathbb{P}^2$ if (and only if) the matrix on the right-hand side has rank $1$ (that is, all its $2 \times 2$ minors vanish). Next, consider the quiver

and consider the representation $V=(V_1,V_2,V_3)$ with vertex-spaces $V_1=\mathbb{C}$, $V_2 = \mathbb{C}^{10}$ and $V_2 = \mathbb{C}^6$. The linear maps $x,y$ and $z$ correspond to the columns of the matrix above, that is

$$(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9) \begin{cases} \rightarrow^x~(x_0,x_1,x_2,x_3,x_4,x_5) \\ \rightarrow^y~(x_1,x_3,x_4,x_6,x_7,x_8) \\ \rightarrow^z~(x_2,x_4,x_5,x_7,x_8,x_9) \end{cases}$$

The linear map $h~:~\mathbb{C}^{10} \rightarrow \mathbb{C}$ encodes the equation of the hyper-plane, that is $h=x_0-x_7+x_9$.

Now consider the quiver Grassmannian $QGr(m,V)$ for the dimension vector $m=(0,1,1)$. A base-vector $p=(x_0,\cdots,x_9)$ of $W_2 = \mathbb{C}p$ of a subrepresentation $W=(0,W_2,W_3) \subset V$ must be such that $h(x)=0$, that is, $p$ determines a point of the hyper-plane.

Likewise the vectors $x(p),y(p)$ and $z(p)$ must all lie in the one-dimensional space $W_3 = \mathbb{C}$, that is, the right-hand side matrix above must have rank one and hence $p$ is a point in the image of $\mathbb{P}^2$ under the Veronese.

That is, $Gr(m,V)$ is isomorphic to the intersection of this image with the hyper-plane and hence is isomorphic to the elliptic curve.

The general case is similar as one can view any projective subvariety $X \hookrightarrow \mathbb{P}^n$ as isomorphic to the intersection of the image of a specific $d$-uple Veronese embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ with a number of hyper-planes in $\mathbb{P}^N$.

le petit village de l’Ariège

For me this quest is over. All i did was following breadcrumbs left by others.

Fellow-travelers arrived there before. What did they do next?

The people from the esoteric site L’Astrée, write literary texts on Grothendieck, mixing strange details (such as the kiosque de la place Pinel, the village of Fougax-et-Barrineuf and even ‘Winnie’ or ‘Fred le Belge, notre indic vers Grothendieck’) with genuine finds, such as this ‘petite annonce’ in the journal for this le 09

which reads:

“RETRAITE (PROFESSEUR UNIVERSITE) CHERCHE -eau de vie de pays pour mes préparations de plantes. Ecrire à M. Grothendieck.”

Caterine Aira makes a movie

Most of you will be perfectly happy to know Grothendieck lives in a tiny village close to the market-town of Saint-Girons. A few may click through the map below to satisfy their need to know the name of ‘le petit village de l’Ariège’.

To do what exactly, i wonder.

You can write a letter, but it will be returned unopened.

You can email ‘la Mairie’ (btw. it’s the ‘orange’-address rather than the ‘wanadoo’ ones), but i doubt they’ll update their Wikipedia-page to acknowledge Grothendieck among the ‘Personnalités liées à la commune’.

You can go there in person to hear the villagers out, but, until you’re a ‘résident permanent’, you will be considered an outsider, and treated as one.

If it’s knowledge you’re after, Grothendieck made it plain he no longer wants to be part of the mathematical society.

His mathematical brain is scattered in the 20.000 pages, kept in 5 boxes at the university of Montpellier. This is the genuine treasure, and should be made public without further delay.

I trust you’ll proceed wisely.

To ‘Monsieur Alexandre’, on his 85th birthday:
happier days!

Previous in this series:
un petit village de l’Ariège

G-spots : Saint-Girons

Roy Lisker (remember him from the Mormoiron post?) has written up his Grothendieck-quest(s), available for just 23$, and with this strange blurb-text:

“The author organized a committee to search for him that led to his discovery, in good health and busily at work, in September, 1996. This committee has since become the Grothendieck Biography Project. All of this is recorded in a 300 page account in 3 parts.”

Probably he refers to the trip made by Leila Schneps and Pierre Lochak, nicely described in Sam Leith’s The Einstein of maths:

“One of the last members of the mathematical establishment to come into contact with him was Leila Schneps. Through a series of coincidences, she and her future husband, Pierre Lochak, learned from a market trader in the town he left in 1991 that ‘the crazy mathematician’ had turned up in another town in the Pyrenees. Schneps and Lochak in due course staked out the marketplace of the town, carrying an out-of-date photograph of Grothendieck, and waited for the greatest mathematician of the 20th century to show up in search of beansprouts.

‘We spent all morning there in the market. And then there he was.’ Were they not worried he’d run away? ‘We were scared. We didn’t know what would happen. But he was really, really nice. He said he didn’t want to be found, but he was friendly. We told him that one of his conjectures had been proved. He had no idea. He’d stopped being interested in maths at that stage. He thought his unpublished work would all have been long forgotten.’”

To city-cats this may seem an improbable coincidence, but if you live in the French mountains for some time, you learn to group your shoppings, and do them on market-days. The nearest market-town, where you can find a decent ‘boulangerie’ or supermarket, may be just 20 kms down the road, but it’ll take you close to an hour to get there.

If you sit near the town-fountain on market-days, for some weeks, you will have seen most of the people living in the vast neighborhood.

So, we’d better try to find Leila’s market-town.

One of the nicer talks on the life of Grothendieck was given by Winfried Scharlau (who also has two books on offer on Grothendieck’s life, seems to become an emerging bisiness …) at the IHES Grothendieck colloque.

Colloque Grothendieck Winfried Scharlau par Ihes_science

This video is stuffed with unknown (at least to me) pictures of Grothendieck, his places at Mormoiron and Villecun and of his four children still living in France. Highly recommended!

But, the lecture has a very, very strange ending.

At 1hr 06.51 into the video he shows the slide reproduced on the left below and says: “Okay and here’s a picture on which I will not further comment. That’s the last thing I want to show you. I thank you very much for your patience.”

Leila Schneps has a page with pictures on her website, including 3 pictures of her house, and then the one on the right above, merely described as ‘Another house’.

And then there’s this paragraph from Roy Lisker’s (him again) Travelogue-France (March 8-April 5, 2005) part 2

“I left the IHP around 11 to return to the CNRS research center at 175 rue du Chevaleret. Pierre Lochak and I discussed the possibility of my going to the town of St. Giron outside of Toulouse to make another impromptu visit to La Maison d’Alexandre Grothendieck.”

So, here we have three founding members of the Grothendieck circle linking publicly to the same picture of that one place they want to keep secret at all cost?

Dream on!

If you followed this series at all and have looked at the pictures of Grothendieck’s houses in Mormoiron or Villecun it is hard to imagine him living in a bourgeois-house, dating from the end of the 19th century, in a medium-sized market-town.

Still, it is quite likely that the picture is indeed taken in Saint-Girons, on some saturday in 1996 when Leila and Pierre bumped into Grothendieck on the market in Saint-Girons.

After all, Saint-Girons is the market-town closest to the final Grothendieck-spot…

Previous in this series:
un petit village de l’Ariège

G-spots : un petit village de l’Ariège

We would love to conclude this series by finding the location of the “final” Grothendieck-spot, before his 85th birthday, this thursday.

But, the road ahead will be treacherous, with imaginary villages along the way and some other traps planted by the nice people of the Grothendieck Fan Club

It is well-known that some members (if not all) of the GFC know the exact location of Grothendieck’s hideout in the Pyrenees. Trying to pry this information from them, pledging to keep the name secret, is described as ‘solving an equation in n unknowns’ in the article Le trésor oublié du génie des maths (h/t +David Roberts):

“Cela fait aujourd’hui vingt-deux ans qu’il vit reclus au pied des Pyrénées, dans un village où personne ne va par hasard et dont le nom doit rester secret. Il le souhaite et ceux qui, de loin, le protègent le souhaitent également. Obtenir l’adresse contre l’assurance de ne pas le déranger prend le temps de résoudre une équation à «n» inconnues. Se poster devant chez lui permet de constater qu’il est bien vivant au milieu d’un village qui le regarde comme «le savant» sans chercher à en savoir plus. A 84 ans, il vient se chauffer au soleil devant son portail puis rentre dans sa maison où nul ne pénètre.”

As we don’t want to take this vow of secrecy, we will have to rely on the few hints they left in the literature. Presumably, the most trustworthy information is to be found in Pierre Cartier’s paper A country of which nothing is known but the name, Grothendieck and “motives”:

“As I already said, he retired in 1988, and has lived since then in self-imposed exile. At first he lived near the Fontaine de Vaucluse, in the middle of a little vineyard that he cultivated, and near to his daughter Johanna and his grandchildren. But later he broke off every family relation. He didn’t seem to mind that the place where he lived was located so near to the infamous Camp du Vernet which played a sad role in his childhood. He lived for years without any contact with the outside world and only a few people even knew where he was. He chose to live alone, considered by his neighbors as a “retired mathematics professor who’s a bit mad”.”

There is a small (but for our purposes important) addition to the first sentence in the French version:

“… il a pris sa retraite en 1988, et vit depuis un exil intérieur dans un petit village de l’Ariège.”

This addition makes our quest a bit more ‘doable’. The department of l’Ariège is one of the lesser populated ones in France (having less that 150.000 inhabitants), and has ‘only’ 332 villages.

One can divide this number roughly by 2, leaving out the larger villages and towns and those situated in the higher mountains, where living must be extremely difficult for an 85 year old.

An alternative reason for leaving out the more southern villages is Cartier’s claim that ‘le petit village’ is close to the Camp du Vernet, which is the place from which Grothendieck’s father was deported to Auschwitz.

This former concentration camp is located in Le Vernet, close to the town of Pamiers (central upper part of the map).

So, one can safely assume that the final G-spot must lie on the map below (click on it to navigate and explore).

Previous in this series:

G-spots : Olmet-et-Villecun

Before we start the quest for the final G-spot, hopefully in time for Grothendieck’s 85th birthday, one more post on Alexandre’s ‘hippy-days’.

In the second part of Allyn Jackson’s “The Life of Alexandre Grothendieck” she tells the story that AG, while touring the US to spread the gospel of the eco-mouvement “Survivre et Vivre” (the deal was that he gave 1 math-talk if he was allowed to give another one on ecology/politics), met a graduate student of Daniel Gorenstein, Justine Skalba, who quickly became a G-groupie and returned with him after the US-trip to France, where she lived with him for two years (and had one child with him, John, who later also became a mathematician).

Allyn Jackson writes:

“In early 1973 he (AG) and Skalba moved to Olmet-le-sec (probably she means: Olmet sec, so without any additions), a rural village in the south of France. This area was at the time a magnet for hippies and others in the counterculture movement who wanted to return to a simpler lifestyle close at hand (I would have added: and, it still is). Here Grothendieck again attempted (he did this once before in his Parisian period, setting up a commune in Chatenay-Malabry) to start up a commune, but personality conflicts led to its collapse. At various times three of Grothendieck’s children came to live in the Paris commune and in the one in Olmet (probably this being: Johanna, Mathieu and Alexandre who even today maintain an alternative lifestyle). After the commune disolved, he moved with Skalba and his children to Villecum, a short distance away.”

As Yves Ladegaillerie tells Jackson, Grothendieck lived an ascetic, unconventional life in an old house without electricity in Villecun, about thirty-five miles outside of Montpellier. Ladegaillerie remembered seeing Justine Skalba and her baby there. Many friends, acquantances and students went to visit Grothendieck there, including people from the ecology movement.

Here’s the (in)famous house in Villecun (h/t Winfried Scharlau)

And, if you are a bit like me, wanting to see everything with G-earth or maps, here’s the scenery (click on the image to be there).

Again, if someone at the Mairie d’Olmet-et-Villecun reads this, please consider adding to your list of ‘Personnalités liées à la commune’

– Michel Chevalier
– Paul Dardé

this one:

Alexandre Grothendieck

Merci infiniment!

Previous in this series:


lieven le bruyn's (old) blog