A p-typical cover of a connected scheme on which p=0 is a finite etale cover whose monodromy group (i.e., the Galois group of its normal closure) is a p-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the p-typical quotients of the etale fundamental groups, and a decomposition theorem for p-typical covers of polynomial rings over an algebraically closed field.
We consider the space $\mathcal R_{g,S_3}^{S_3}$ of curves with a connected $S_3$-cover, proving that for any odd genus $g\geq 13$ this moduli is of general type. Furthermore we develop a set of tools that are essential in approaching the case of $G$-covers for any finite group $G$.
Given a triple cover p: X --> Y of varieties, we produce a new variety Z and a birational morphism f: Z --> X which is an isomorphism away from the fat-point ramification locus of p. The variety Z has a natural interpretation in terms of the data describing the triple cover, and the morphism f has an elegant geometric description.
The tempered fundamental group of a p-adic analytic space classifies coverings that are dominated by a topological covering (for the Berkovich topology) of a finite etale covering of the space. Here we construct cospecialization homomorphisms between (p') versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log reduction. In particular, we will have to study invariance of the geometric log fundamental group of…
This paper constructs cospecialization homomorphisms between the (p') versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction (the tempered fundamental group is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). We studied the question for families of curves in another paper. To construct them, we will start by describing the pro-(p') tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tem…
In this article, examples of Zariski pairs $(B_1, B_2)$ satisfying the following condition are given: (i) $°B_1 = °B_2 = 7$. (ii) Irreducible components of $B_i$ $(i = 1, 2)$ are lines and conics. (iii) Singularities of $B_i$ $(i = 1, 2)$ are nodes, tacnodes and ordinary triple points. In order to construct $B_i$ ($i = 1, 2$), we make use of geometry of sections of rational elliptic surfaces and their group structure. Dihedral covers play important roles to distinguish the topology of $({\mathbb P}^2, B_i)$ $(i = 1, 2)$.
This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry.
In a recent paper Chiodo and Farkas described the singular locus and the locus of non-canonical singularities of the moduli space of level curves. In this work we generalize their results to the moduli space $\overline{\mathcal R}_{g,G}$ of curves with a $G$-cover for any finite group $G$. We show that non-canonical singularities are of two types: $T$-curves, that is singularities lifted from the moduli space $\overline{\mathcal M}_g$ of stable curves, and $J$-curves, that is new singularities entirely characterized by the dual graph of the cover. Finally, we prove that in the case $G=S_3$, th…
A nilpotent orbit $O$ of a complex semisimple Lie algebra $\mathfrak{g}$ has finite fundamental group. Associated with an etale cover of $O$, we have a finite cover of the closure $\bar{O}$ of $O$. In this article we consider the finite cover X associated with the universal cover of a nilpotent orbit $O$ of a classical simple Lie algebra $\mathfrak{g}$. We construct explicitly a Q-factorial terminalization of X in a group theoretic way.
We study algebro-geometric properties of determinantal loci of (n+1)th symmetric matrices and also their double covers for even ranks. Their singularities, Fano indices and birational geometries are studied in general. The double covers of symmetric determinantal loci of rank four are studied with special interest by noting their relation to the Hilbert schemes of conics on Grassmannians.
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
Lecture notes of an algebraic geometry graduate course. The topics covered are as follows. Cohomology: ext sheaves and groups, cohomology with support, local cohomology, local duality. Duality: relative duality, Cohen-Macaulay schemes. Morphisms of schemes: flatness and base change and related topics. Hilbert and Quotient schemes.
In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the 'etale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.
The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.
In this work we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form, but can rather contain fibral divisors or multiple sections (i.e. a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this we employ well established tools from the mathematics li…
In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov-Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian su…
We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP^3 branched along singular quartic surfaces. On these threefolds we find families of real lines in explicit form and prove that which families have to be chosen as twistor lines depend on how we take small resolutions of the double covering. This is a first step for determining t…
These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may stimulate readers to investigate them.
The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises is attached. Our notes are intended for the graduate students and faculty with interests in noncommutative geometry; they can be read by non-experts in the field.
More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $f\colon \mathbb{A}^n\to \mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use kn…
