Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGraphs, $mathbb{F}_1$-schemes and virtual mixed Tate motives
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In a number of recent works [6, 7] the authors have introduced and studied a functor $mathcal{F}_k$ which associates to each loose graph $Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that $mathcal{F}_k(Gamma)$ is largely controlled by the combinatorics of $Gamma$. Here, $k$ is a field, and we allow $k$ to be $mathbb{F}_1$, the field with one element. For each finite prime field $mathbb{F}_p$, it is noted in [6] that any $mathcal{F}_k(Gamma)$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we show that for each $k$, the class of $mathcal{F}_k(Gamma)$ in the Grothendieck ring $K_0(texttt{Sch}_k)$ is contained in $mathbb{Z}[mathbb{L}]$, the integral subring generated by the virtual Lefschetz motive.
rate research
Read More
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdans motivic integration and Ayoubs equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
In [19] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $mathbb{F}_1$) to a so-called loose graph (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over $mathbb{F}_1$ such as combinatorial $mathbb{F}_1$-projective and $mathbb{F}_1$-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit., and show that Deitmar schemes which are defined by finite trees (with possible end points) are defined over $mathbb{F}_1$ in Kurokawas sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated $mathbb{F}_q$-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Iharas zeta function (which is trivial in this case). Using a process called surgery, we show that one can determine the zeta function of a general loose graph and its associated { Deitmar, Grothendieck }-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of $mathbb{F}_1$-schemes along the way in order to perform the calculations. Finally, we compare the new zeta function to Iharas zeta function for graphs in a number of examples, and include a computer program for performing more tedious calculations.
We express nested Hilbert schemes of points and curves on a smooth projective surface as virtual resolutions of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
In this essay we study various notions of projective space (and other schemes) over $mathbb{F}_{1^ell}$, with $mathbb{F}_1$ denoting the field with one element. Our leading motivation is the Hiden Points Principle, which shows a huge deviation between the set of rational points as closed points defined over $mathbb{F}_{1^ell}$, and the set of rational points defined as morphisms $texttt{Spec}(mathbb{F}_{1^ell}) mapsto mathcal{X}$. We also introduce, in the same vein as Kurokawa [13], schemes of $mathbb{F}_{1^ell}$-type, and consider their zeta functions.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
