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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 separated noetherian base. Combined with a theorem of Deligne on the existence of enough points, this provides an algebro-geometric description of a conservative family of fibre functors on these categories of sheaves. As an example of an application we show direct image along a closed immersion is exact for all these topologies except qfh. The methods are transportable to other categories of sheaves as well.
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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 recently, it was shown that one can always find nonspecial curves through the expected number of general points and linear spaces. After some expository material regarding scrolls, we consider the generalization of this question to varieties of all dimensions and explain why smooth varieties of minimal degree satisfy interpolation. We give twenty-two equivalent formulations of interpolation. We also classify when Castelnuovo curves satisfy weak interpolation. In the appendix, we prove that del Pezzo surfaces satisfy weak interpolation. Our techniques for proving interpolation include deformation theory, degeneration and specialization, and association.
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.
We here present rudiments of an approach to geometric actions in noncommutative algebraic geometry, based on geometrically admissible actions of monoidal categories. This generalizes the usual (co)module algebras over Hopf algebras which provide affine examples. We introduce a compatibility of monoidal actions and localizations which is a distributive law. There are satisfactory notions of equivariant objects, noncommutative fiber bundles and quotients in this setup.
We introduce and describe the $2$-category $mathsf{Grt}_{flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $boxtimes$ restricts nicely to $mathsf{Grt}_{flat}$. Then, we characterize exponentiable objects with respect to $boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.
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 Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry.
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