Course Details for A.Y. 2018/2019

Course Objectives

The course intends to introduce the students to the area of algebraic Geometry. This is done by starting with plane algebraic curves, precisely conics and cubics e given their classification. Then one considers the relation between Algebra and Geometry presenting the Ideal-Variety correspondence and also given a geometric interpretation of the operations between ideals of polynomials. One considers both the affine and projective case. In particular the students become familiar with some classical varieties, as the twisted cubic, the Veronese variety, the Segre variety and with some aspects of the geometry of quadric hypersurfaces.

Course Content

• PLANE ALGEBRAIC CURVES Examples of parametrized curves. Parametrization of a plane conic. Homogeneous polynomial in two variables. Bézout's Theorem (easy cases). The space of all conics. Conics passing thru at most 5 points. Intersection of two conics. Degenerate conics in a pencil of conics.
• Plane cubics and their classification Examples of parametrized cubics. The cubic y^2=x(x-1)(x-k) does not have a rational parametrization. Linear systems of curves. Cubics throughout 8 points The space of all plane cubics and its dimension. Pascal's theorem. Local properties of plane algebraic curves. Intersection multiplicity of a plane curve in a point. Notion of singular point of a plane curve. Inflexion points of a plane curve. The Hessian of a plane curve. Notion of principal tangents. How to compute the principal tangents of a plane curve. Preliminaries for the classification of plane cubics. Salmon's theorem. Notion of module of a plane cubic. Projective equivalence classes of non singular plane cubics. Classification of non singular plane cubics. Classification of irreducible singular plane cubics. Classification of singular plane cubics.
• VARIETIES IN THE AFFINE SPACE. Noetherian Rings: Definition. Example. The Hilbert bases theorem. Algebraic sets. Zariski's topology. Examples of algebraic sets. Irreducible algebraic sets. Notion of k-algebra; k-algebra of finite type. Correspondence algebraic sets- ideals. Operations with ideals and their geometric meaning. Nullstellensatz: Radical Ideals. Hilbert's Nullstellensatz Theorem: weak form and strong form. Applications of the Hilbert's Nullstellensatz Theorem.
• Affine Varieties: Coordinates ring. Polynomial functions. Polynomial maps. Isomorphisms. Properties of Polynomial maps. Rational maps on affine varieties The file of the rational functions. Domain of a rational function. Ideal of the denominator of a rational function. Rational maps. Composition of rational maps. Induced homomorphism. Birational map. Dominant rational map. Morphisms. Standard open subsets.
• VARIETIES IN THE PROJECTIVE SPACE Homogeneous ideals. The homogenous correspondence. The proiettive Hilbert's Nullstellensatz. Rational functions on a variety. Regular functions. Affine covering of a projective variety. Rational maps and morphisms: Rational maps with values in an affine variety; Rational maps with values in an projective variety; Regular maps; Morphisms. Birational maps. Rational varieties. Segre Variety Segre map. Notion of Segre Variety . Subvarieties of the Segre Variety. Twisted cubic. Equations of the twisted cubic in P^1xP^1. Veronese Variety Veronese map. Notion of Veronese Variety. Veronese surface . Examples of varieties birational to P^2 but not isomorphic to P^2. Subvarieties of the Veronese Varietye. Combining Verones maps and Segre maps.
• Tangent Space Tangent Space to a hyper surface. Notion of non singular /singular point of an algebraic variety. Tangent Space to an affine variety. Zariski's Tangent space. Notion of degree of a variety. Degree of some varieties: hyper surface; normal rational cubic; Veronese variety; Segre variety S_{2,1}. Blowup map. The blow-up of a point in the affine space. The blow-up of a point in the projective plane. Quadric Hypersurfaces: The geometry of the quadric hypersurfaces. Plucker coordinate of a line of P^3. The variety of line of P^3 is isomorphic to a quadric in P^5. Caracterization of smooth quadric in P^n. Linear spaces contained in a non singular quadric.

Assessment Methods and Criteria

Oral exam

Textbooks

• M.L. Fania, Appunti del docente a.a.2017-2018    
• A. Biancofiore, Appunti di Geometria Algebrica, a.a. 2015/2016    
• E. Sernesi, Geometria 1 , Bollati Boringhieri . 2000.    
• D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms , Springer. 2007. Undergraduate Texts in Mathematics    
• J. Harris, Lectures on Algebraic Geometry,Graduate Texts in Mathematics , Springer Verlag. (vol. 133) 1992.    
• M. Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts , Cambridge University Press . (vol. 12) 1988.    

Course page updates

• A.Y. 2017/2018
• A.Y. 2018/2019