This note covers the following topics: Hochschild cohomology and group actions, Differential Weil Descent and Differentially Large Fields, Minimum positive entropy of complex Enriques surface automorphisms, Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces, Superstring Field Theory, Superforms and Supergeometry, Picard groups for tropical toric schemes, Complex multiplication and Brauer groups of K3 surfaces.
Author(s): arXiv.org
This note covers the following topics: Functors, Isomorphic and equivalent categories, Representable functors, Some constructions in the light of representable functors, Schemes: Definition and basic properties, Properties of morphisms of schemes, general techniques and constructions.
Author(s): Audun Holme
This book is intended to give a serious and reasonably complete introduction to algebraic geometry, not just for experts in the field. Topics covered includes: Sheaves, Schemes, Morphisms of schemes, Useful classes of morphisms of schemes, Closed embeddings and related notions, Fibered products of schemes, and base change, Geometric properties: Dimension and smoothness, Quasicoherent sheaves, Quasicoherent sheaves on projective A-schemes, Differentials,Derived functors, Power series and the Theorem on Formal Functions, Proof of Serre duality.
Author(s): Ravi Vakil
This book explains the following topics: Systems of algebraic equations, Affine algebraic sets, Morphisms of affine algebraic varieties, Irreducible algebraic sets and rational functions, Projective algebraic varieties, Morphisms of projective algebraic varieties, Quasi-projective algebraic sets, The image of a projective algebraic set, Finite regular maps, Dimension, Lines on hypersurfaces, Tangent space, Local parameters, Projective embeddings and Riemann-Roch Theorem.
Author(s): Igor V. Dolgachev
This book covers the following topics: Introduction and Motivation, General definitions and results, Cubic curves, Curves of higher genus.
Author(s): MattKerr
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Author(s): NA
This is an introductory course note in algebraic geometry. The author has trodden lightly through the theory and concentrated more on examples.
Author(s): Donu Arapura
Combinatorics and Algebraic Geometry have classically enjoyed a fruitful interplay. The aim of this series of lectures is to introduce recent development in this research area. The topics involve classical algebraic varieties endowed with a rich combinatorial structure, such as toric and tropical varieties.
Author(s): Giorgio Ottaviani
This note contains the following subtopics of Algebraic Geometry, Theory of Equations, Analytic Geometry, Affine Varieties and Hilbert’s Nullstellensatz , Projective Varieties and Bezout’s Theorem, Epilogue
Author(s): Sudhir R. Ghorpade
An introduction to both the geometry and the arithmetic of abelian varieties. It includes a discussion of the theorems of Honda and Tate concerning abelian varieties over finite fields and the paper of Faltings in which he proves Mordell's Conjecture. Warning: These notes are less polished than the others.
Author(s): J.S.Milne
This note contains the following subtopics: Basics of commutative algebra, Affine geometry, Projective geometry, Local geometry, Divisors.
Author(s): Alexei Skorobogatov
This note covers the following topics: Cohomology, Relative duality, Properties of morphisms of schemes, Cohen-Macaulay schemes, Hilbert and Quotient schemes.
Author(s): Caucher Birkar
This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Covered topics are: Basics of category theory, Sheaves, Abelian sheaves, Schemes, Morphisms of schemes, Sheaves of modules, More properties of morphisms, Projective morphisms, Projective morphisms, Flat morphisms and descent, Differentials Divisors, Divisors on curves, Homological algebra, Sheaf cohomology, Cohomology of quasicoherent sheaves, Cohomology of projective spaces, Hilbert polynomials, GAGA, Serre duality for projective space, Dualizing sheaves and RiemannRoch, CohenMacaulay schemes and Serre duality, Higher RiemannRoch and Etale cohomology.
Author(s): Prof. Kiran Kedlaya
These notes are for a first graduate course on algebraic geometry. It is assumed that the students are not familiar with algebraic geometry. Author has taken a moderate approach emphasising both geometrical and algebraic thinking.
Author(s): Caucher Birkar
This note covers the following topics: The Pre-cursor of Bezout’s Theorem: High School Algebra, The Projective Plane and Homogenization, Bezout’s Theorem and Some Examples.
Author(s): Stephanie Fitchett, Florida Atlantic University Honors College
This note covers the following topics: The correspondence between ideals and algebraic sets, Projections, Sheaves, Morphisms of Sheaves, Glueing Sheaves, More on Spec(R), Proj(R)is a scheme, Properties of schemes, Sheaves of modules, Schemes over a field, sheaf of differentials and Picard group.
Author(s): Sandra Di Rocco
This book explains the following topics: Etale Morphisms, Etale Fundamental Group, The Local Ring for the Etale Topology, Sheaves for the Etale Topology, Direct and Inverse Images of Sheaves, Cohomology: Definition and the Basic Properties, Cohomology of Curves, Cohomological Dimension, Purity; the Gysin Sequence, The Proper Base Change Theorem, Cohomology Groups with Compact Support, The Smooth Base Change Theorem, The Comparison Theorem, The Kunneth Formula, Proof of the Weil Conjectures, The Weil Conjectures, The Geometry of Lefschetz Pencils and Cohomology of Lefschetz Pencils.
Author(s): J.S. Milne
It is hoped that this note will assist students in untangling the morass: they approach the subject from what could cynically be described as a rather narrow perspective, but they contain far more than the usual amount of detail and they include simple examples illustrating how algebraic geometers would work within this limited context.
Author(s): R.C. Churchill
This book explains the following topics: Polarity, Conics, Plane cubics, Determinantal equations, Theta characteristics, Plane Quartics, Planar Cremona transformations, Del Pezzo surfaces, Cubic surfaces, Geometry of Lines.
Author(s): Igor V. Dolgachev
These notes are an introduction to the theory of algebraic varieties. In contrast to most such accounts they study abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory.
Author(s): J.S. Milne
The material presented here consists of a more or less self contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, is to understand the Enriques classification of surfaces from the point of view of Mori theory.
Author(s): Chris Peters
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Author(s): NA
This note explains the following topics: Affine Varieties, Hilbert’s Nullstell, Projective and Abstract Varieties, Grassmann varieties and vector bundles, Finite morphisms, Dimension Theory, Regular and singular points, Tangent space, Complete local rings, Intersection theory.
Author(s): Yuriy Drozd
This is an introductory course note in algebraic geometry. Author has trodden lightly through the theory and concentrated more on examples.Covered topics are: Affine Geometry, Projective Geometry, The category of varieties, Dimension theory and Differential calculus.
Author(s): Donu Arapura
This book explains the following topics: The geometry of monoids, Log structures and charts, Morphisms of log schemes, Differentials and smoothness, De Rham and Betti cohomology.
Author(s): Arthur Ogus
This note explains the theory of (complex) algebraic surfaces, with the goal of understanding Enriques' classification of surfaces.
Author(s): Ravi Vakil
This note explains the following topics: Affine and projective curves: algebraic aspects, Affine and projective curves: topological aspects.
Author(s): M.J. de la Puente
This note explains the following topics: Algebraic surfaces, Singularities, Maximal numbers of singularities, Quartics, Enumerative geometry.
Author(s): Bruce Hunt
This book covers the following topics: Elementary Algebraic Geometry, Dimension, Local Theory, Projective Geometry, Affine Schemes and Schemes in General, Tangent and Normal Bundles, Cohomology, Proper Schemes and Morphisms, Sheaves and Ringed Spaces.
Author(s): Jean Gallier
This note describes the relation between algebraic curves and Riemann surfaces.
Author(s): Joel W. Robbin
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