# Algebraic Geometry and Arithmetic Curves

@inproceedings{Liu2002AlgebraicGA, title={Algebraic Geometry and Arithmetic Curves}, author={Qing Liu}, year={2002} }

Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of differentials 7. Divisors and applications to curves 8. Birational geometry of surfaces 9. Regular surfaces 10. Reduction of algebraic curves Bibilography Index

## 1,065 Citations

SHEAVES SCHEMES AND THEIR APPLICATIONS

- Mathematics
- 2011

Chapter 1 Illustrates some concepts of Commutative algebra.
Chapter 2 Gives a basic introduction to Sheaf theory.
Chapter 3 Illustrates the Cohomology theory of Sheaves.
Chapter 4 Deals with…

Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

- Mathematics
- 2008

We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for…

Complete intersections: moduli, Torelli, and good reduction

- Mathematics
- 2015

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved…

Specialization of linear systems from curves to graphs

- Mathematics, Computer Science
- 2007

The interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface is investigated and some applications to graph theory, arithmetic geometry, and tropical geometry are provided.

Explicit arithmetic intersection theory and computation of Néron-Tate heights

- Mathematics, Computer ScienceMath. Comput.
- 2020

The algorithm is implemented, and it is shown how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares.

On the Geometry of Global Function Fields, the Riemann–Roch Theorem, and Finiteness Properties of S-Arithmetic Groups

- Mathematics
- 2012

In this survey I sketch Behr–Harder reduction theory for reductive groups over global functions fields and briefly describe its applicability in the theory of S-arithmetic groups, notably homological…

Fields of definition and Belyi type theorems for curves and surfaces

- Mathematics
- 2016

We study the relationship between the (effective) fields of definition of a complex projective variety and the orbit {X}σ∈Aut(C) where X is the “twisted” variety obtained by applying σ to the…

Cohomological invariants of algebraic stacks

- Mathematics
- 2014

The purpose of this paper is to lay the foundations of a theory of invariants in etale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and…

Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks

- Mathematics
- 2011

This survey discusses hyperbolicity properties of moduli stacks and generalisations of the Shafarevich Hyperbolicity Conjecture to higher dimensions. It concentrates on methods and results that…