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We develop the framework for augmented homotopical algebraic geometry. This is an extension of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. To do so, we define the notion of augmentation categories, which are a special class of generalised Reedy categories. For an augmentation category, we prove the existence of a closed Quillen model structure on the presheaf category which is compatible with the Kan-Quillen model structure on simplicial sets. Moreover, we use the concept of augmented hypercovers to define a local model structure on the category of augmented presheaves. We prove that crossed simplicial groups, and the planar rooted tree category are examples of augmentation categories. Finally, we introduce a method for generating new examples from old via a categorical pushout construction.
This is an expanded version of the two papers Interpolation of Varieties of Minimal Degree and Interpolation Problems: Del Pezzo Surfaces. It is well known that one can find a rational normal curve in $mathbb P^n$ through $n+3$ general points. More r
We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, etale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separ
We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hau
A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential ge
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling geometric constrai