Reineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects over the elusive field with one element F1.
In a comment to the previous post Markus Reineke recalls motivating discussions with Javier Lopez Pena and Oliver Lorscheid (the guys responsable for the map of F1-land above) and asks about potential connections with F1-geometry. In this post I will ellaborate on Javier’s response.
The Kapranov-Smirnov F1-floklore tells us that an n-dimensional vectorspace over F1 is a pointed set V∙ consisting of n+1 points, the distinguished point playing the role of the zero-vector. Linear maps V∙→W∙ between F1-spaces are then just maps of pointed sets (sending the distinguished element of V∙ to that of W∙). As an example, the base-change group GLn(F1) of an n-dimensional F1-space V∙ is isomorphic to the symmetric group Sn.
This allows us to make sense of quiver-representations over F1. To each vertex we associate a pointed set and to each arrow a map of pointed sets between the vertex-pointed sets. The dimension-vector α of quiver-representation is defined as before and two representations with the same dimension-vector are isomorphic is they lie in the same orbit under the action of the product of the symmetric groups determined by the components of α. All this (and a bit more) has been worked out by Matt Szczesny in the paper Representations of quivers over F1.
Oliver Lorscheid developed his own approach to F1 based on the notion of blueprints (see also part 2 and a paper with Javier).
Roughly speaking a blueprint B=A//R is a commutative monoid A together with an equivalence relation R on the monoid semiring N[A] compatible with addition and multiplication. Any commutative ring R is a blueprint by taking A the multiplicative monoid of R and R(∑iai,∑jbj) if and only if the elements ∑iai and ∑jbj in R are equal.
One can extend the usual notions of prime ideals, Zariski topology and structure sheaf from commutative rings to blueprints and hence define a notion of “blue schemes” which are then taken to be the schemes over F1.
What’s the connection with Reineke’s result? Well, for quiver-representations V defined over F1 they can show that the corresponding quiver Grassmannians Gr(V,α) are blue projective varieties and hence are geometric objects defined over F1.
For us, old-fashioned representation theorists, a complex quiver-representation V is defined over F1 if and only if there is an isomorphic representation V′ with the property that all its arrow-matrices have at most one 1 in every column, and zeroes elsewhere.
Remember from last time that Reineke’s representation consisted of two parts : the Veronese-part encoding the d-uple embedding Pn→PM and a linear part describing the subvariety X→Pn as the intersection of the image of Pn in PM with a finite number of hyper-planes in PM.
We have seen that the Veronese-part is always defined over F1, compatible with the fact that all approaches to F1-geometry allow for projective spaces and d-uple embeddings. The linear part does not have to be defined over F1 in general, but we can look at the varieties we get when we force the linear-part matrices to be of the correct form.
For example, by modifying the map h of last time to h=x0+x7+x9 we get that the quiver-representation
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is defined over F1 and hence that Reineke’s associated quiver Grassmannian, which is the smooth plane elliptic curve V(x3+y2z+z3), is a blue variety. This in sharp contrast with other approaches to F1-geometry which do not allow elliptic curves!
Oliver will give a talk at the 6th European Congress of Mathematics in the mini-symposium Absolute Arithmetic and F1-Geometry. Judging from his abstract,he will also mention quiver Grassmannians.
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