Related papers: Artin-Schreier-Witt Theory for Stacky Curves
We classify stacky curves in characteristic $p > 0$ with cyclic stabilizers of order $p$ using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a…
Let $K=k((t))$ be a local field of characteristic $p>0$, with perfect residue field $k$. Let $\vec{a}=(a_0,a_1,\dots,a_{n-1})\in W_n(K)$ be a Witt vector of length $n$. Artin-Schreier-Witt theory associates to $\vec{a}$ a cyclic extension…
The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and…
The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin-Schreier extensions. We define refined versions of invariants of…
We present an algorithm which, given a connected smooth projective curve $X$ over an algebraically closed field of characteristic $p>0$ and its Hasse--Witt matrix, as well as a positive integer $n$, computes all \'etale Galois covers of $X$…
We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads…
We analyse the diagonal quotient for products of certain Artin--Schreier curves. The smooth models are almost always surfaces of general type, with Chern slopes tending asymptotically to 1. The calculation of numerical invariants relies on…
We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite…
In this paper we study the semi-stable reduction of $p$ and $p^2$-cyclic covers of curves in equal characteristic $p>0$. The main tool we use is the classical Artin-Schreier-Witt theory for $p^n$-cyclic covers in characteristic $p$.…
We study the canonical model of the Artin-Schreier-Kummer-Witt flat family of curves over a ring of mixed characteristic. We first prove the relative version of a classical theorem by Petri, then use the model proposed by Bertin-M\'ezard to…
Given a discrete valued field $K$ of positive characteristic, we study the cyclic lifting problem of purely inseparable extensions of the residue field. We prove that unlike the mixed characteristic case, cyclic lifts of any finite purely…
We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to L\^e and Teissier, which reformulates Whitney regularity in terms of conormal…
We present an algorithmic framework for computing generators for the ring of invariants of an Artin-Schreier curve. We give explicit invariants for almost all Artin-Schreier curves of genus up to~8 in standard form, and for a handful of…
We introduce equivariant Chow-Witt groups in order to define Chow-Witt groups of quotient stacks. We compute the Chow-Witt ring of the moduli stack of stable (resp. smooth) elliptic curves, providing a geometric interpretation of the new…
We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety.…
In local class field theory, the Schmid-Witt symbol encodes interesting data about the ramification theory of $p$-extensi-ons of $K$ and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid…
We develop a theory for stable maps to curves with divisible ramification. For a fixed integer $r>0$, we show that the condition of every ramification locus being divisible by $r$ is equivalent to the existence of an $r$th root of a…
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on…
M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in…
The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the…