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Grothendieck’s gribouillis

Published January 3, 2015 by lievenlb

A math-story well worth following in 2015.

What will happen to Grothendieck’s unpublished notes, or as he preferred, his “gribouillis” (scribbles)?

Here’s the little I know about this:

1. The Mormoiron scribbles

During the 80ties Grothendieck lived in ‘Les Aumettes’ in Mormoiron

In 1991, just before he moved to the Pyrenees he burned almost all of his personal notes in the garden. He phoned Jean Malgoire:

“Si tu ne viens pas chercher mon bordel mathématique, il va brûler avec le reste.”

Malgoire sped to Mormoiron and rescued 5 boxes containing about 20.000 pages. The next 20 years he kept them in his office, not exactly knowing what to do with them.

On january 3rd 2010, Grothendieck wrote his (in)famous letter forbidding others to share or publish any of his writings. (Picture via the SecretBloggingSeminar)

Malgoire feared that Grothendieck would soon ask him to destroy the Mormoiron-gribouillis and decided to donate them to the University of Montpellier.

They are kept somewhere in their archives, the exact location known only to Jean Malgoire, Luc Gomel (who is in charge of the patrimonium of the University of Montpellier) and the person who put the boxes away.

After Grothendieck’s death on november 13th, FranceTV3 did broadcast a short news-item.

If Grothendieck’s children agree, the University of Montpellier intends to make an inventory of the 5 boxes and will make them available, at least to researchers.

2. The Lasserre scribbles

The final 23 years of his life, Grothendieck lived in the small village of Lasserre in the French Pyrenees.

Here he could be glimpsed blurrily through the window as he wrote for hours during the night.(Picture via the GrohendieckCircle)

Leila Schneps and her husband Pierre Lochak did visit the house and met with Grothendieck’s family the week after his death.

Before she went, she was optimistic about the outcome as she emailed:

“I have already started modifying the Grothendieck circle website and it will of course eventually return completely. Plus many things will be added, as we will now have access to Grothendieck’s correspondence and many other papers.”

Her latest comment, from december 16th, left on the Grothendieck-circle bulletin board, is more pessimistic:

“Il a ecrit a Lasserre sans cesse pendant plus de 20 ans. Je n’ai pu que jeter un rapide coup d’oeil sur tout ce qu’il a laisse. Il y a de tout: des maths, des reflexions sur lui-meme, et des reflexions sur la nature humaine et sur l’univers. Rien n’est disponible pour le moment. Ces manuscrits finiront dans une bibliotheque et seront peut-etre un jour consultables.”

The good news is that there appears to be some mathematics among the Lassere-gribouillis. The sad news being that none of this is available at the moment, and perhaps never will.

So, what happened? Here’s my best guess:

Grothendieck’s children were pretty upset a private letter from one of them turned up in the French press, a couple of years ago.

Perhaps, they first want to make sure family related material is recovered, before they’ll consider donating the rest (hopefully to the University of Montpellier to be reunited with Grothendieck’s Mormoiron-notes).

This may take some time.

Further reading (in French):

Grothendieck, mon tresor (nationale)

Un génie mystérieux, un secret bien gardé

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$\mathbf{Ext}(\mathbb{Q},\mathbb{Z})$ and the solenoid $\widehat{\mathbb{Q}}$

Published July 31, 2014 by lievenlb

Note to self: check Jack Morava’s arXiv notes on a more regular basis!

It started with the G+-post below by +David Roberts:

Suddenly I realised I hadn’t checked out Morava‘s “short preprints with ambitious ideas, but no proofs” lately.

A couple of years ago I had a brief email exchange with him on the Habiro topology on the roots of unity, and, in the process he send me a 3 page draft with ideas on how this could be relevant to higher dimensional topological QFT (If my memory doesn’t fail me, I can’t find anything remotely related in the arXiv-list).

Being in a number-theory phase lately (yes, I also have to give next year, for the first time, in the second semester, a master-course on Number Theory) the paper A topological group of extensions of $\mathbb{Q}$ by $\mathbb{Z}$ caught my eyes.

The extension group $Ext(\mathbb{Q},\mathbb{Z})$ classifies all short exact sequences of Abelian groups

$0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Q} \rightarrow 0$

upto equivalence, that is commuting sequences with end-maps being identities.

The note by Boardman Some Common Tor and Ext Groups hs a subsection on this group/rational vector space, starting out like this:

“This subsection is strictly optional. The group $Ext(\mathbb{Q}, \mathbb{Z})$ is much more difficult to determine. It is easy to see that it is a rational vector space, simply from the presence of $\mathbb{Q}$, but harder to see what its dimension is. This group is not as mysterious as is sometimes claimed, but is related to adèle groups familiar to number theorists.”

Boardman goes on to show that this extension group can be identified with $\mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q}$ where $\mathbb{A}^f_{\mathbb{Q}}$ is the ring of finite adèles, that is, sequence $(x_2,x_3,x_5,…)$ of $p$-adic numbers $x_p \in \widehat{\mathbb{Q}}_p$ with all but finitely many $x_p \in \widehat{\mathbb{Z}}_p$, and $\mathbb{Q}$ is the additive subgroup of constant sequences $(x,x,x,…)$.

Usually though, one considers the full adèle ring $\mathbb{A}_{\mathbb{Q}} = \mathbb{R} \times \mathbb{A}^f_{\mathbb{Q}}$ and one might ask for a similar interpretation of the adèle class-group $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$.

This group is known to be isomorphic to the character group (or Pontrtrjagin dual) of the rational numbers, that is, to $\widehat{\mathbb{Q}}$ which are all group-morphisms $\mathbb{Q} \rightarrow S^1$ from the rational numbers to the unit circle. This group is sometimes called the ‘solenoid’ $\Sigma$, it is connected but not path connected and the path-component of the identity $\Sigma_0 = \mathbb{R}$.

A very nice and accessible account of the solenoid is given in the paper The character group of $\mathbb{Q}$ by Keith Conrad.

The point of Morava’s note is that he identifies the solenoid $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}$ with a larger group of ‘rigidified’ extensions $Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z})$.That is, one starts with a usual extension in $Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$ as above, but in addition, one fixes a splitting of the induced sequence

$0 \rightarrow \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow A \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow \mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{R} \rightarrow 0$

Forgetting the splitting this gives the exact sequence

$0 \rightarrow \mathbb{R} \rightarrow Ext_{\mathbb{Z}_0}(\mathbb{Q},\mathbb{Z}) \rightarrow Ext_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z}) \rightarrow 0$

which is isomorphic to the sequence involving the path-component of the solenoid!

$0 \rightarrow \Sigma_0 = \mathbb{R} \rightarrow \Sigma=\widehat{Q} \rightarrow \mathbb{A}^f_{\mathbb{Q}}/\mathbb{Q} \rightarrow 0$

Morava ends with: “I suppose the proposition above has a natural reformulation
in Arakelov geometry; but I don’t know anything about Arakelov geometry”…

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On categories, go and the book $\in$

Published July 24, 2014 by lievenlb

A nice interview with Jacques Roubaud (the guy responsible for Bourbaki’s death announcement) in the courtyard of the ENS. He talks about go, categories, the composition of his book $\in$ and, of course, Grothendieck and Bourbaki.

Clearly there are pop-math books like dedicated to $\pi$ or $e$, but I don’t know just one novel having as its title a single mathematical symbol : $\in$ by Jacques Roubaud, which appeared in 1967.

The book consists of 361 small texts, 180 for the white stones and 181 for the black stones in a game of go, between Masami Shinohara (8th dan) and Mitsuo Takei (2nd Kyu). Here’s the game:

In the interview, Roubaud tells that go became quite popular in the mid sixties among French mathematicians, or at least those in the circle of Chevalley, who discovered the game in Japan and became a go-envangelist on his return to Paris.

In the preface to $\in$, the reader is invited to read it in a variety of possible ways. Either by paying attention to certain groupings of stones on the board, the corresponding texts sharing a common theme. Or, by reading them in order of how the go-game evolved (the numbering of white and black stones is not the same as the texts appearing in the book, fortunately there’s a conversion table on pages 153-155).

Or you can read them by paragraph, and each paragraph has as its title a mathematical symbol. We have $\in$, $\supset$, $\Box$, Hilbert’s $\tau$ and an imagined symbol ‘Symbole de la réflexion’, which are two mirrored and overlapping $\in$’s. For more information, thereader should consult the “Dictionnaire de la langue mathématique” by Lachatre and … Grothendieck.

According to the ‘bibliographie’ below it is number 17 in the ‘Publications of the L.I.T’.

Other ‘odd’ books in the list are: Bourbaki’s book on set theory, the thesis of Jean Benabou (who is responsible for Roubaud’s conversion from solving the exercises in Bourbaki to doing work in category theory. Roubaud also claims in the interview that category theory inspired him in the composition of the book $\in$) and there’s also Guillaume d’Ockham’s ‘Summa logicae’…

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A noncommutative moduli space

Published July 19, 2014 by lievenlb

Supernatural numbers also appear in noncommutative geometry via James Glimm’s characterisation of a class of simple $C^*$-algebras, the UHF-algebras.

A uniformly hyperfine (or, UHF) algebra $A$ is a $C^*$-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras

$M_{c_1}(\mathbb{C}) \subset M_{c_2}(\mathbb{C}) \subset … \quad \subset A$

Such embedding are only possible if the matrix-sizes divide each other, that is $c_1 | c_2 | c_3 | … $, and we can assign to $A$ the supernatural number $s=\prod_i c_i$ and denote $A=A(s)$.

In his paper On a certain class of operator algebras, Glimm proved that two UHF-algebras $A(s)$ and $B(t)$ are isomorphic as $C^*$-algebras if and only if $s=t$. That is, the supernatural numbers $\mathbb{S}$ are precisely the isomorphism classes of UHF-algebras.

An important invariant, the Grothendieck group $K_0$ of $A(s)$, can be described as the additive subgroup $\mathbb{Q}(s)$ of $\mathbb{Q}$ generated by all fractions of the form $\frac{1}{n}$ where $n$ is a positive integer dividing $s$.

A “noncommutative space” is a Morita class of $C^*$-algebras, so we want to know when two $UHF$-algebras $A(s)$ and $B(t)$ are Morita-equivalent. This turns out to be the case when there are positive integers $n$ and $m$ such that $n.s = m.t$, or equivalently when the $K_0$’s $\mathbb{Q}(s)$ and $\mathbb{Q}(t)$ are isomorphic as additive subgroups of $\mathbb{Q}$.

Thus Morita-equivalence defines an equivalence relation on $\mathbb{S}$ as follows: if $s=\prod p^{s_p}$ and $t= \prod p^{t_p}$ then $s \sim t$ if and only if the following two properties are satisfied:

(1): $s_p = \infty$ iff $t_p= \infty$, and

(2): $s_p=t_p$ for all but finitely many primes $p$.

That is, we can view the equivalence classes $\mathbb{S}/\sim$ as the moduli space of noncommutative spaces associated to UHF-algebras!

Now, the equivalence relation is described in terms of isomorphism classes of additive subgroups of the rationals, which was precisely the characterisation of isomorphism classes of points in the arithmetic site, that is, the finite adèle classes

$\mathbb{S}/\sim~\simeq~\mathbb{Q}^* \backslash \mathbb{A}^f_{\mathbb{Q}} / \widehat{\mathbb{Z}}^*$

and as the induced topology of $\mathbb{A}^f_{\mathbb{Q}}$ on it is trivial, this “space” is usually thought of as a noncommutative space.

That is, $\mathbb{S}/\sim$ is a noncommutative moduli space of noncommutative spaces defined by UHF-algebras.

The finite integers form one equivalence class, corresponding to the fact that the finite dimensional UHF-algebras $M_n(\mathbb{C})$ are all Morita-equivalent to $\mathbb{C}$, or a bit more pompous, that the Brauer group $Br(\mathbb{C})$ is trivial.

Multiplication of supernaturals induces a well defined multiplication on equivalence classes, and, with that multiplication we can view $\mathbb{S}/\sim$ as the ‘Brauer-monoid’ $Br_{\infty}(\mathbb{C})$ of simple UHF-algebras…

(Btw. the photo of James Glimm above was taken by George Bergman in 1972)

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Oulipo’s use of the Tohoku paper

Published July 17, 2014 by lievenlb

Many identify the ‘Tohoku Mathematical Journal’ with just one paper published in it, affectionately called the Tohoku paper: “Sur quelques points d’algèbre homologique” by Alexander Grothendieck.

In this paper, Grothendieck reshaped homological algebra for Abelian categories, extending the setting of Cartan-Eilenberg (their book and the paper both appeared in 1957). While working on the Tohoku paper in Kansas, Grothendieck did not have access to the manuscript of the 1956 book of Cartan-Eilenberg, about which he heard from his correspondence with Serre.

Concerning the title, an interesting suggestion was made by Mathieu Bélanger in his thesis “Grothendieck et les topos: rupture et continuité dans les modes d’analyse du concept d’espace topologique”, (footnote 18 on page 164):

“There is a striking resemblance between the title of the Grothendieck’s article “Sur quelques points d’algèbre homologique”, and that of Fréchet‘s thesis “Sur quelques points d’analyse fonctionelle”. Why? Grothendieck remains silent about it. Perhaps he saw a methodological similarity between the introduction, by Fréchet, of abstract spaces in order to develop the foundations of functional calculus and that of the Abelian categories he needed to clarify the homological theory. Compared with categories of sets, groups, topological spaces, etc. that were used until then, Abelian categories are in effect abstract categories.”

But, what does this have to do with the literary group OuLiPo (ouvroir de littérature potentielle, ‘workshop of potential literature’)?

Oulipo was founded in 1960 by Raymond Queneau and François Le Lionnais. Other notable members have included novelists Georges Perec and Italo Calvino, poets Oskar Pastior, Jean Lescure and poet/mathematician Jacques Roubaud.

Several members of Oulipo were either active mathematicians or at least had an interest in mathematics. Sometimes, Oulipo is said to be the literary answer to Bourbaki. The group explored new ways to create literature, often with methods coming from mathematics or programming.

One such method is described in “Chimères” by Le Lionnais:

One takes a source text A. One ’empties’ it, that is, one deletes all nouns, adjectives and verbs, but marks where they were in the text. In this way we have ‘prepared’ the text.

Next we take three target texts and make lists of words from them, K the list of nouns of the first, L the list of adjectives of the second and M the list of verbs of the third. Finally, we fill the empty spaces in the source text by words from the target lists, in the order that they appeared in the target texts.

In the example Le Lionnais gives, the liste M is the list of all verbs appearing in the Tohoku paper.


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Grothendieck’s Café

Published July 1, 2014 by lievenlb

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”. At the time only an undefined idea of “schéma” was current in Paris, meaning more or less whatever would improve on Weil’s foundations.” (McLarty in The Rising Sea)

Finding that particular café in Paris, presumably in the 5th arrondissement, seemed like looking for a needle in a haystack.

Until now.

In trying to solve the next riddle in Bourbaki’s death announcement:

A reception will be held at the Bar ‘The Direct Products’, at the crossroads of the Projective Resolutions (formerly Koszul square)

I’ve been reading Mathematics, a novel by Jacques Roubaud (the guy responsible for the announcement) on Parisian math-life in the 50ties and 60ties.

It turns out that the poor Bourbakistas had very little choice if they wanted to have a beer (or coffee) after attending a seminar at the IHP.

On page 114, Roubaud writes:

“Père Plantin presided over his bar, which presided over the Lhomond/Ulm street corner. It is an obvious choice. rue Pierre-et-Marie-Curie had no bars; rue d’Ulm had no bars in eyeshot either. If we emerged, as we did, on this side of the Institut Henri Poincaré (for doing so on the other side would have meant fraternizing with the Spanish and Geography students in the cafés on rue Saint-Jacques, which was out of the question), we had no choice. Café Plantin had a hegemony.”

It is unclear to me whether Plantin was once actually the name of the café, or that it’s just Roubaud’s code-word for it. At other places in the book, e.g. on pages 82 and 113, he consistently writes “Plantin”, between quotes.

Today, the café on the crossroads of rue d’Ulm (where the Ecole Normal Superieure is located) and de rue Lhomond is the Interlude Café

and here’s what Roubaud has to say about it, or rather about the situation in 1997, when the French version of his book was published:

“the thing that would currently be found at the very same corner of rues Lhomond/Ulm would not be what I am here terming “Plantin”.”

So, we can only hope that the Café ‘Aux Produits Directs’ was a lot cosier, way back then.

But let us return to Grothendieck’s “What is a scheme?” story.

Now that we have a fair idea of location, what about a possible date? Here’s a suggestion: this happened on monday december 12th, 1955, and, one of the friends present must have been Cartier.

Here’s why.

The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.

The lecturer was Claude Chevalley, and the date was december 12th 1955.


I’m fairly certain Grothendieck and Cartier attended and that Cartier was either briefed before or understood the stuff at once (btw. he gave another talk on schemes, a year later at the Chevalley seminar).

A couple of days later, on december 15th, Grothendieck sends a letter to his pal Serre (who must have been out of Paris for otherwise they’d phone each other) ending with:



Note the phrase: I am exploiting him most profitably. Yes, by asking him daft questions over a pint at Café “Plantin”…

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the birthday of Grothendieck topologies

Published June 23, 2014 by lievenlb

This is the story of the day the notion of ‘neighbourhood’ changed forever (at least in the geometric sense).

For ages a neighbourhood of a point was understood to be an open set of the topology containing that point. But on that day, it was demonstrated that the topology of choice of algebraic geometry, the Zariski topology, needed a drastic upgrade.

This ultimately led to the totally new notion of Grothendieck topologies, which aren’t topological spaces at all.

Formally, the definition of Grothendieck topologies was cooked up in the fall of 1961 when Grothendieck visited Zariski, Mike Artin and David Mumford in Harvard.

The following spring, Mike Artin ran a seminar resulting in his lecture notes on, yes, Grothendieck topologies.

But, paradigm shifts like this need a spark, ‘une bougie d’allumage’, and that moment of insight happened quite a few years earlier.

It was a sunny spring monday afternoon at the Ecole Normal Superieure. Jean-Pierre Serre was giving the first lecture in the 1958 Seminaire Claude Chevalley which that year had Chow rings as its topic.

That day, april 21st 1958, Serre was lecturing on algebraic fibre bundles:

He had run into a problem.

If a Lie group $G$ acts freely on a manifold $M$, then the set of $G$-orbits $M/G$ is again a manifold and the quotient map $\pi : M \rightarrow M/G$ is a principal $G$-fibre bundle meaning that for sufficiently small open sets $U$ of $M/G$ we have diffeomorphisms

$\pi^{-1}(U) \simeq U \times G$

that is, locally (but not globally) $M$ is just a product manifold of $G$ with another manifold and the $G$-orbits are all of the form $\{ u \} \times G$.

The corresponding situation in algebraic geometry would be this: a nice, say reductive, algebraic group $G$ acting freely on a nice, say smooth, algebraic variety $X$. In this case one can form again an orbit space $X/G$ which is again a (smooth) algebraic variety but the natural quotient map $\pi : X \rightarrow X/G$ rarely has this local product property…

The reason being that the Zariski topology on $X/G$ is way too coarse, it doesn’t have enough open sets to enforce this local product property.

(For algebraists: let $A$ be an Azumaya algebra of rank $n^2$ over $\mathbb{C}[X]$, then the representation variety $\mathbf{rep}_n(A)$ is a principal $\mathbf{PGL}_n$-bundle over $X$ but is only local trivial in the Zariski topology when $A$ is a trivial Azumaya algebra, that is, $End_{\mathbb{C}[X]}(P)$ for a rank $n$ projective module $P$ over $\mathbb{C}[X]$.)

But, Serre had come up with a solution.

He was going to study fibre bundles which were locally ‘isotrivial’, meaning that they had the required local product property but only after extending them over an unamified cover $Y \rightarrow X$ (what we now call, an etale cover) and he was able to clasify such fibre bundles by a laborious way (which we now call the first etale cohomology group).

The story goes that Grothendieck, sitting in the public, immediately saw that these etale extensions were the correct generalization of the usual (Zariski) localizations and that he could develop a cohomology theory out of them in all dimensions.

According to Colin McLarty Serre was ‘absolutely unconvinced’, since he felt he had ‘brutally forced’ the bundles to yield the $H^1$’s.

We will never known what Serre actually wrote on the blackboard on april 21st 1958.

The above scanned image tells it is an expanded version of the original talk, written up several months later after the ICM-talk by Grothendieck in Edinburgh.

By that time, Grothendieck had shown Serre that his method indeed gives cohomology in all dimensions,and convinced him that this etale cohomology was likely to be the “true cohomology needed to prove the Weil conjectures”.

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European mathematics in 1927

Published June 20, 2014 by lievenlb

Here’s a map of the (major) mathematical centers in Europe (in 1927), made for the Rockefeller Foundation.



Support by the Rockefeller foundation was important for European Mathematics between the two world wars. They supported the erection of the Mathematical Institute in Goettingen between 1926-1929 and creation of the Institut Henri Poincare in Paris at about the same time.

Careers of people such as Stefan Banach, Bartel van der Waerden and Andre Weil benefitted hugely from becoming fellows of the Rockefeller-funded International Educational Board in the 20ties.

The map itself shows that there were three major centers at the time: Goettingen, Paris and Rome (followed by Berlin and Oxford, at a distance).

Also the distribution by topics (the pie-charts per university) is interesting: predominantly Analysis (red) with a fair share of Geometry (yellow), Number Theory (green) and Applied Mathematics (blue). Philosophy (black) was even more important than Algebra (orange) which existed only in Goettingen (Noether, van der Waerden) and Berlin.

I’d love to see a similar map for 2014…

A larger version of the map can be found here.

There’s a corresponding map for the USA here.

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Quiver Grassmannians can be anything

Published May 2, 2013 by lievenlb

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 $\mathbb{P}^2 \rightarrow \mathbb{P}^9$ 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 $X \rightarrow \mathbb{P}^n$ as isomorphic to the intersection of the image of a specific $d$-uple Veronese embedding $\mathbb{P}^n \rightarrow \mathbb{P}^N$ with a number of hyper-planes in $\mathbb{P}^N$.

ADDED For those desperate to read the original comments-section, here’s the link.

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