Contents:
Oct 6, 2011– Lecture 1
Introduction.Definition of (affine) algebraic set.Examples.
([1] Chapter 0, examples 0.5-0.7,0.13-0.15)
Oct 10, 2011 – Lecture 2
Maps V and I and their properties. Examples.
([1] Chapter 1, §1.1 and [3] ex.1.6)
Oct 10, 2011 – Lecture 3
Radical ideals.Hilbert’s Nullstellensatz.
Irreducible algebraic sets and prime ideals.
([1] Chapter 1, §1.1)
Oct 13, 2011 – Lecture 4
Examples.Hypersurfaces and principal ideals.
([1] Chapter 1, §1.1, and [2] §3.11-worked examples)
Decomposition of an algebraic set in irreducible components.
([1] Chapter 1, §1.1 and [4] Chapter 1, §1)
Oct 17, 2011 – Lecture 5
Definition of affine variety.Zariski topology.Topological dimension of an algebraic set.
Properties of Zariski topology.Product of algebraic sets.Topology of the products.
([1] Chapter 1, §1.1 and [4] Chapter 1, §1)
Oct 17, 2011 – Lecture 6
Polynomial functions and maps: coordinate ring of an affine variety. Examples
([1] Chapter 1, §1.2.1)
Oct20, 2011 – Lecture 7
Polynomial maps and k-algebra homomorphisms. Examples
([1] Chapter 1, §1.2.2)
Functor between the category of affine varieties and of f.g., reduced, k-algebras.
([1] Chapter 1, §1.2.3).
Oct 24, 2011 – Lecture 8
Product of irreducible algebraic sets ([5] pg. 35 or [1] Chapter 1, §1.2.4)
Rational functions and maps.
([1] Chapter 1, §1.3)
Oct 24, 2011 – Lecture 9
Dominant rational maps and homorphisms between function fields.Examples.
([1] Chapter 1, §1.3.3)
Quasi-Affine varieties and birational maps.
([1] Chapter 1, §1.3.4)
Oct 27, 2011 – Lecture 10
Examples of rational affine varieties: cubic curve with a cusp, quadric surface and
cubic surface with two skew lines.
([5] §3.3, examples 1 and 2)
Isomorphism between a quasi-affine variety and an affine variety.
([1] Chapter 1, §1.3.4)
Nov 2, 2011 – Lecture 11
Isomorphism between a quasi-affine variety and an affine variety.
([1] Chapter 1, §1.3.4)
Projective spaces, duality, and projective sets.
([1] Chapter 2, §2.1 and 2.2)
Nov 3, 2011 – Lecture 12
Examples of projective varieties: the rational normal cubic and the variety of its chords.
([6] Lecture 1)
The Segre embedding of P1 xP1.
([6] Lecture 2)
Nov 7, 2011 – Lecture 13
P5 parameterizing the conics of P2: the cubic hypersurface of singular conics
and lines corresponding to pencils of conics. The Veronese surface.
([6] Lecture 4, example 4.8)
Nov 7, 2011 – Lecture 14
Graduate rings and homogeneous ideals.
([1] Chapter 2, §2.1 and 2.2)
Nov 10, 2011 – Lecture 15
Affine cones.Projective Hilbert’s Nullstellensatz. Projective algebraic varieties.
Zariski topology on Pn.
([1] Chapter 2, §2.1 and 2.2)
Nov 14, 2011 – Lecture 16
Rational functions on projective varieties and morphisms
([1] Chapter 2,§2.3)
Nov 14, 2011 – Lecture 17
Rational and birational maps.Examples.
([1] Chapter 2,§2.3, [5]Chapter 1,§4.4)
Nov 16, 2011 – Lecture 18
Product of projective varieties and Segre embedding.Projection maps.
([5]Chapter 1,§5.1)
Nov 21, 2011 – Lecture 19
Subvarieties of the Segre varieties.
([5] Chapter 1, §5.2)
Categorical products and fiber products.
([6] Chapter 2)
Nov 21, 2011– Lecture 20
Graph of a regular map.
Image of a projective variety under regular maps.
([5] Chapter 1, §5.2)
Nov 23, 2011 – Lecture 21
Finite maps between affine varieties and their properties.
Finite maps between projective varieties.
([5]Chapter 1,§5.3)
Nov 24, 2011 – Lecture 22
Projections and Noether Normalization Lemma.
([5]Chapter 1,§5.4)
Projections of projective curves and surfaces.
Nov 28, 2011– Lecture 23
Graph of a rational map. Birational maps
([6]Lecture 7)
Blows-up.
([1] Chapter 2, §2.3.6)
Nov 28, 2011– Lecture 24
Examples: resolution of singularities of plane curves, quadratic transformations
and blow-up of a quadric in one point.
([1] Chapter 2, §2.3.6, [4] Chapter 5, example 4.2.3)
Nov 30, 2011– Lecture 25
Blow-up of a quadric in one point.
Unirational varieties.
([6]Lecture 7)
Smooth points and tangent spaces to hypersurfaces and to affine varieties.
([1] Chapter 3, §3.1)
Dec1, 2011 – Lecture 26
Dimension of a variety through tangent spaces.
([1] Chapter 3, §3.1)
Dec5, 2011 – Lecture 27
Algebraic Characterization of the dimension of a variety and behavior of
dimension under birational maps.
([1] Chapter 3, §3.2)
Dec5, 2011 – Lecture 28
Krull dimension and equivalence between all the above definitions of dimension.
([4] Chapter 1 and [1] Chapter 3, §3.2)
Dimension of the intersection of a projective variety with a hypersurface.
Dec12, 2011 – Lecture 29
Dimension of the intersection of projective varieties.
Dimension of fibres of a regular map.
([5]Chapter 1,§6.2,§6.3)
Dec12, 2011 – Lecture 30
Geometric definition of dimension of a projective variety
([6] Lecture 11)
Geometric definition of degree of a projective variety and Bezout Theorems.
([6] Lecture 7)
Dec14, 2011 – Lecture 31
Introduction to Grassmmann varieties.
Definition of G(1,3).([7])
Dec15, 2011 – Lecture 32
Subvarieties of G(1,3), linear complexes.
Dec19, 2011 – Lecture 33 and 34
Rationality of the quadratic complex.
Examples of enumerative geometry ([7])
Dec21, 2011 – Lecture 35
Lines on a generic surface of P^3.
([5] Chapter 1 - §6.4)
Dec22, 2011 – Lecture 36
Lines on a generic cubic surface of P^3.
([5] Chapter 1 - §6.4)
Definition of G(k,n).
Intrinsic definition of the Plucker embedding.([5])
Jan9, 2012 – Lecture 37
Introduction to maps to projective spaces.
Vector Bundles and Line bundles. Sections of a vector bundle. Examples of Vector Bundles on algebraic varieties.
([8] Chapter 8 - §8.1, §8.2,§8.3 )
Jan9, 2012 – Lecture 38
Tautological line bundle and Hyperplane bundle on a projective variety and their
global sections.
([8] Chapter 8 - §8.4)
Jan11, 2012 – Lecture 39
Universal bundle on Grassmannians. Pull-back of a vector bundle. Dual of a vector bundle.
Tangent and cotangent bundle on a projective variety and canonical line bundle.
([8] Chapter 8 - §8.4)
Jan16, 2012 – Lecture 40 and 41
Line Bundles and rational maps. Examples. Spanned and Very ample Line Bundles.
Overview on canonical embbeddings of projective smooth curves.
([8] Chapter 8 - §8.5)
References:
[1] K.Hulek – Elementary Algebraic Geometry – AMS
[2] M.Reid – Undergraduate Algebraic Geometry – London Mathematical Society Student Texts 12
[3] K.Ueno–Algebraic Geometry 1 – From Algebraic Varieties to Schemes – Translations of Mathematical Monographs – AMS Vol. 185.
[4] R. Hartshorne Algebraic geometry Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
[5] I. Shafarevich Basic algebraic geometry vol. 1 Springer-Verlag, New York-Heidelberg, 1977.
[6] J. Harris Algebraic geometry (a first course) Graduate Texts in Math. No. 133.Springer-Verlag, New York-Heidelberg, 1977.
[7] S. L. Kleiman and Dan Laksov Schubert Calculus The American Mathematical Monthly, Vol. 79, No. 10 (Dec., 1972), pp. 1061-1082PDF
[8]K.E.Smith-L.Kahanpaa-p.Kekalainen-W.Traves–An Invitation to Algebraic geometry –Universitext Texts –Springer.