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Register a new userAutomorphisms of Deitmar schemes, I. Functoriality and Trees
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In a recent paper [3], the authors introduced a map $mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $mathbb{F}_1$) with any given graph $Gamma$. By base extension, a scheme $mathcal{X}_k = mathcal{F}(Gamma) otimes_{mathbb{F}_1} k$ over any field $k$ arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the schemes $mathcal{X}_k$. Several automorphism groups are considered: combinatorial, topological, and scheme-theoretic groups, and also groups induced by automorphisms of the ambient projective space. When $Gamma$ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.
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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 provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar schemes with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine automorphism groups of the Deitmar schemes and base extensions to fields.
The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederers result about Grobner strata of zero-dimensional ideals, proving that Grobner strata of any ideals are locally closed subschemes of Hilbert schemes.
We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-Hubsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-Hubsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.
This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $mathscr{G}$ of homological dimension $le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its derived category and verify the conjecture in various cases. This framework allows us to relax certain regularity conditions on various known formulae -- such as the ones for blowups (along Koszul-regular centers), Cayleys trick, standard flips, projectivizations, and Grassmannain-flips -- and supplement these formulae with the results on mutations and relative Serre functors. This framework also leads us to many new phenomena such as virtual flips, and structural results for the derived categories of (i) $mathrm{Quot}_2$ schemes, (ii) flips from partial desingularizations of $mathrm{rank}le 2$ degeneracy loci, and (iii) blowups along determinantal subschemes of codimension $le 4$.
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