Contents:
Sept 25, 2014 – Lectures 1 and 2
Introduction. Definition of (affine) algebraic set.
Examples: skew cubic curve, GL(n,K) as algebraic sets.
([1] Chapter 0, examples 0.5-0.7, 0.13-0.15)
Maps V and I and their properties. Examples.
([1] Chapter 1, §1.1 and [3] ex.1.6)
Sept 29, 2014 – Lectures 3 and 4
Radical ideals. Hilbert’s Nullstellensatz.
Definition of Zariski topology.
Irreducible algebraic sets and prime ideals. Definition of affine variety.
The skew cubic curve is irreducible.
([1] Chapter 1, §1.1)
Oct 2, 2014 – Lectures 5 and 6
Example of reducible algebraic set.
Decomposition of an algebraic set in irreducible components.
([1] Chapter 1, §1.1, [2] §3.11-worked examples)
Properties of the Zariski topology . Product of algebraic sets. Topology of the products.
Topological dimension of an algebraic set.
([1] Chapter 1, §1.1 and [4] Chapter 1, §1)
Oct 6, 2014 – Lectures 7 and 8
Polynomial functions and maps: coordinate ring of an affine variety. Examples
([1] Chapter 1, §1.2.1)
Rational functions and quotient field of K[V].
([1] Chapter 1, §1.3)
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 9, 2014 – Lectures 9 and 10
Rational maps. Dominant rational maps and homorphisms between function fields. Examples.
([1] Chapter 1, §1.3.3)
Birational maps. ([1] Chapter 1, §1.3.4) and isomorphism between function fields.
Oct 13, 2014 – Lectures 11 and 12
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 principal open set and an affine variety.
([1] Chapter 1, §1.3.4)
Projective spaces, duality, and projective sets.
([1] Chapter 2, §2.1 and 2.2
Oct 16, 2014 – Lectures 13 and 14
Examples of projective varieties: the rational normal cubic.
([6] Lecture 1)
The variety of the chords to the rational normal cubic.
([6] Lecture 1)
The Segre embedding of P1 xP1.
([6] Lecture 2)
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)
Oct 20, 2014 – Lectures 15 and 16
Properties of Veronese surface.
([6] Lecture 4, example 4.8)
Spaces parameterizing hyperquadrics and Veronese maps.
Graduate rings and homogeneous ideals.
([1] Chapter 2, §2.1 and 2.2)
Oct 23, 2014 – Lectures 17 and 18
Affine cones. Projective Hilbert’s Nullstellensatz. Projective algebraic varieties.
Zariski topology on Pn.
([1] Chapter 2, §2.1 and 2.2)
Rational functions on projective varieties and morphisms
([1] Chapter 2, §2.3).
Oct 27, 2014 – Lecture 19 and 20
Rational functions on projective varieties and morphisms
([1] Chapter 2, §2.3)
Rational and birational maps. Examples.
([1] Chapter 2, §2.3, [5] Chapter 1, §4.4)
Oct 30, 2014 – Lecture 21 and 22
Product of projective varieties and Segre embedding. Projection maps.
([5] Chapter 1, §5.1)
Subvarieties of the Segre varieties.
([5] Chapter 1, §5.2)
Segre products as categorical products.
([6] Chapter 2)
Nov 3, 2014 – Lecture 23 and 24
Graph of a regular map.
Image of a projective variety under regular maps.
([5] Chapter 1, §5.2)
Introduction to Blow-up of A^2 in the origin.
Nov 6, 2014 – Lecture 25 and 26
Blows-up.
([1] Chapter 2, §2.3.6)
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)
Rational and Unirational projective varieties.
([6] Lecture 7)
Nov 10, 2014 – Lecture 27 and 28
Finite maps between affine varieties and their properties (with finite fibers,
surjective, closed). Examples.
Finite maps between projective varieties.
([5] Chapter 1, §5.3)
Projections are finite morphisms.
Nov 13, 2014 – Lecture 29 and 30
Noether Normalization Lemma and geometric consequences.
([5] Chapter 1, §5.4)
Smooth points and tangent spaces to hypersurfaces and to affine varieties.
([1] Chapter 3, §3.1)
Nov 17, 2014 – Lecture 31 and 32
Smooth points and tangent spaces to hypersurfaces and to affine varieties.
Dimension of a variety through tangent spaces.
([1] Chapter 3, §3.1)
Algebraic Characterization of the dimension of a variety and behavior of
dimension under birational maps.
([1] Chapter 3, §3.2)
Dimension of the intersection of a projective variety with a hypersurface.
([5] Chapter 1, §6.2, §6.3)
Nov 20, 2014 – Lecture 33 and 34
Dimension of the intersection of projective varieties.
([5] Chapter 1, §6.2, §6.3)
Geometric definition of dimension of a projective variety
([6] Lecture 11)
Geometric definition of degree of a projective variety and Weak Bezout Theorem.
([6] Lecture 7)
Nov 24, 2014 – Lecture 35 and 36
Algebraic Preliminaries to Hilbert Polynomial: primary decomposition of an
homogeneous ideal and examples.
Definition of the Hilbert function of an homogeneous
ideal and of a projective set. Examples: Hilbert function of a set of points.
Properties of the Hilbert function.
([8] chapters 2,3)
Nov 27, 2014 – Lecture 37 and 38
Hilbert polynomial and dimension and degree of a projective variety.
([8] chapters 2,3)
Introduction to Grassmmann varieties.
Definition of G(1,3).([7])
Dec 1, 2014 – Lecture 39 and 40
Subvarieties of G(1,3): linear complexes.
Rationality of the quadratic complex.
Definition of G(k,n).
Examples of enumerative geometry ([7])
Dec 4, 2014 – Lecture 41 and 42
Dimension of fibres of a regular map.
([5] Chapter 1, §6.2, §6.3)
Lines on a generic surface of P^3.
([5] Chapter 1 - §6.4)
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. 1061-1082
[8] E. Arrondo – Introduction to projective varieties – Lecture notes of Phd courses.