Matthew Satriano

Toric stacks I: The theory of stacky fans

The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. In this paper, we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a stacky fan. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stressing stacky phenomena such as canonical stacks and good moduli space morphisms. We also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of P^n and [A^1/G_m]. Indeed, smooth toric stacks precisely solve moduli problems specified by (generalized) effective Cartier divisors with given linear relations and given intersection relations. Smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric DM stacks in papers by Cox and Perroni, respectively. We include a plethora of examples to illustrate the general theory. We hope that this theory of toric stacks can serve as a companion to an introduction to stacks, in much the same way that toric varieties can serve as a companion to an introduction to schemes.

Toric stacks II: Intrinsic characterization of toric stacks

The purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated. In the same spirit, the main result of this paper is that any Artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses.

Torus quotients as global quotients by finite groups

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that

There is no degree map for 0-cycles on Artin stacks

\mathbb{P}^2/A_5

Combining mathematical and statistical modeling to simulate time course bulk and single cell gene expression data in cancer with CancerInSilico

is not expressible as a quotient of a smooth variety by a finite abelian group.

On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (1) the degree of an ordinary point is non-zero, and (1) degrees are compatible with closed immersions.

Logarithmic Geometry and Moduli

Motivation Bioinformatics techniques to analyze time course bulk and single cell omics data are advancing. The absence of a known ground truth of the dynamics of molecular changes challenges benchmarking their performance on real data. Realistic simulated time-course datasets are essential to assess the performance of time course bioinformatics algorithms. Results We develop an R/Bioconductor package CancerInSilico to simulate bulk and single cell transcriptional data from a known ground truth obtained from mathematical models of cellular systems. This package contains a general R infrastructure for cell-based mathematical model, implemented for an off-lattice, cell-center Monte Carlo mathematical model. We also adapt this model to simulate the impact of growth suppression by targeted therapeutics in cancer and benchmark simulations against bulk in vitro experimental data. Sensitivity to parameters is evaluated and used to predict the relative impact of variation in cellular growth parameters and cell types on tumor heterogeneity in therapeutic response. Availability and Implementation CancerInSilico is implemented in an R/Bioconductor package by the same name. Applications presented are available from https://github.com/FertigLab/CancerInSilico-Figures.Bioinformatics techniques to analyze time course bulk and single cell omics data are advancing. The absence of a known ground truth of the dynamics of molecular changes challenges benchmarking their performance on real data. Realistic simulated time-course datasets are essential to assess the performance of time course bioinformatics algorithms. We develop an R/Bioconductor package, CancerInSilico, to simulate bulk and single cell transcriptional data from a known ground truth obtained from mathematical models of cellular systems. This package contains a general R infrastructure for running cell-based models and simulating gene expression data based on the model states. We show how to use this package to simulate a gene expression data set and consequently benchmark analysis methods on this data set with a known ground truth. The package is freely available via Bioconductor: http://bioconductor.org/packages/CancerInSilico/

On a notion of “Galois closure” for extensions of rings

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassmans theorem that we prove in the appendix.