Introduction. Definition of (Affine) Algebraic Set

Contents:

Sept 24, 2015 – 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 28, 2015 – 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) Example of reducible algebraic set.

Oct 1, 2015 – Lectures 5 and 6 Decomposition of an algebraic set in irreducible components. ([1] Chapter 1, §1.1, [2] §3.11-worked examples) Properties of the Zariski topology . Topological dimension of an algebraic set. ([1] Chapter 1, §1.1 and [4] Chapter 1, §1) Polynomial functions and maps: coordinate ring of an affine variety. Examples ([1] Chapter 1, §1.2.1)

Oct 5, 2015 – Lectures 7 and 8 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 12, 2015 – Lectures 9 and 10 Example of extension of a rational function. Product of algebraic sets. Topology of the products. 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 15, 2015 – 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) Properties of Veronese surface. ([6] Lecture 4, example 4.8)

Oct 22, 2015 – Lectures 15 and 16 Spaces parameterizing hyperquadrics and Veronese maps. Graduate rings and homogeneous ideals. ([1] Chapter 2, §2.1 and 2.2) Affine cones. Projective Hilbert’s Nullstellensatz. Projective algebraic varieties. Zariski topology on Pn. ([1] Chapter 2, §2.1 and 2.2)

Oct 26, 2015 – Lectures 17 and 18 Rational functions on projective varieties and morphisms ([1] Chapter 2, §2.3). 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 29, 2015 – Lectures 19 and 20 Product of projective varieties and Segre embedding. Projection maps. ([5] Chapter 1, §5.1) Subvarieties of the Segre varieties. ([5] Chapter 1, §5.2)

Nov 2, 2015 – Lectures 21 and 22 Segre products as categorical products. ([6] Chapter 2) Graph of a regular map. Image of a projective variety under regular maps. ([5] Chapter 1, §5.2)

Nov 5, 2015 – Lectures 23 and 24 Blow-up of A^2 in the origin and generalizations. ([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 9, 2015 – Lecture 25 and 26 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. Noether Normalization Lemma (geometric statement). ([5] Chapter 1, §5.4)

Nov 12, 2015 – Lecture 27 and 28 Noether Normalization Lemma (algebraic statement) 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 23, 2015 – Lecture 29 and 30 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 26, 2015 – Lecture 31 and 32 Transcendence field extensions and definition of transcendence degree. Dimension of the intersection of projective varieties. ([5] Chapter 1, §6.2, §6.3)

Nov 30, 2015 – Lecture 33 and 34 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) Algebraic Preliminaries to Hilbert Polynomial: primary decomposition of an homogeneous ideal and examples.

Dec 3, 2015 – Lecture 35 and 36 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)

Dec 10, 2015 – 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(k,n) and Plucker embedding.([7])

Dec 14, 2015 – Lecture 39 and 40 Ideal of G(k,n). ([7]) Subvarieties of G(1,3): linear complexes. Rationality of the quadratic complex. Examples of enumerative geometry ([7])

Dec 16, 2015 – Lecture 41, 42 and 43

Example of enumerate geometry: Theorem of Clebsch-Salmon. Dimension of fibers of a regular map and irriducibility. ([5] Chapter 1, §6.2, §6.3) Lines on a generic surface of P^3. ([5] Chapter 1 - §6.4) Definition of pre-varieties and (separated) varieties as generalization of affine varieties.

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.