Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? We show there is a scheme M B/A parameterizing the choice of a generator θ ∈ B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.A choice of a generator θ is a point of the scheme M B/A . This inspires a local-to-global study of monogeneity, piecing together monogenerators over points, completions, open sets, and so on. Local generators may not come from global ones, but they often glue to twisted monogenerators that we define. We show a number ring has class number one if and only if each twisted monogenerator is in fact a global generator θ. The moduli spaces of various twisted monogenerators are either a Proj or stack quotient of M B/A by natural symmetries. The various moduli spaces defined can be used to apply cohomological tools and other geometric methods for finding rational points to the classical problem of monogenic algebra extensions.

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category A-mod of modules over a fixed ring A in a topos E. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site A-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by llusie.

Costello's pushforward formula relates virtual fundamental classes of virtually birational algebraic stacks. Its original formulation omits a necessary hypothesis, whose addition is not sufficient to correct the proof. We supply a substitute for Costello's notion of pure degree and prove the pushforward formula with this definition.We also show the hypotheses of the corrected pushforward formula are satisfied in a variety of its applications. Some adjustments to the original proofs are required in several cases, including the original one.

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras B/A, when is B generated by a single element θ ∈ B over A? In this paper, we show there is a scheme M B/A parameterizing the choice of a generator θ ∈ B, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

Let V, W be a pair of smooth varieties. We want to compare curve counts on V × W with those on V and W . The product formula in Gromov-Witten theory compares the virtual fundamental classes of stable maps to a product M g,n (V × W ) to the product of stable maps M g,n (V ) × M g,n (W ). We prove the analogous theorem for log stable maps to log smooth varieties V, W .This extends results of Y.P. Lee and F. Qu, who introduced this formula after K. Behrend. We introduce "log normal cones" and "log virtual fundamental classes," as well as modified versions of standard intersection-theoretic machinery adapted to log geometry.