Lectures on algebraic varieties over \(\mathbb F_1\) (Q2904914)

This paper contains a refinement of the approach to \(\mathbb F_1\)-geometry given by the author in [Mosc. Math. J. 4, No. 1, 217--244 (2004; Zbl 1103.14003)]. He defines an ``affine gadget'' similar to the ``truc'' of loc. cit., the change being that the `point-functor' now is defined on the category of abelian groups rather than rings. Toric varieties as examples of varieties defined over \({\mathbb F}_1\) are studied in some detail. At the end the author gives a construction of an \({\mathbb F}_1\)-variety from a Euclidean lattice which counts the points of absolute value at most one inspired by a construction of \textit{N. Durov} in [``New approach to Arakelov geometry'', \url{arXiv:0704.2030}] and he raises the question whether there is a way to attach to the latter a tariffed variety in the sense of [\textit{J. López Peña} and \textit{O. Lorscheid}, Math. Z. 267, No. 3--4, 605--643 (2011; Zbl 1220.14021)].NEWLINENEWLINEFor the entire collection see [Zbl 1243.14002].