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In order to develop the foundations of logarithmic derived geometry, we introduce a model category of logarithmic simplicial rings and a notion of derived log etale maps and use this to define derived log stacks.
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We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $mathbb{S} to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra.
In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck-Witt group of a derived stack, and compare it to the classical one.
In this short paper we outline (mostly without proofs) our new approach to the derived category of sheaves of commutative DG rings. The proofs will appear in a subsequent paper. Among other things, we explain how to form the derived intersection of two closed subschemes inside a given algebraic scheme X, without recourse to simplicial or higher homotopical methods, and without any global assumptions on X.
We develop the theory of resolvent degree, introduced by Brauer cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilberts 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilberts 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.
This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analyt
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