Prof. Giuseppe Pareschi

Department of Mathematics

Viale della Ricerca Scientifica 1, 00133, Roma, IT

Stanza: 0212

Telefono: 06 72594621

pareschi@mat.uniroma2.it

ALGEBRAIC GEOMETRY, YEAR 2015-'16, LAUREA MAGISTRALE IN MATEMATICA PURA E APPLICATA

Instructor: Prof. Giuseppe Pareschi

Timetable:

Tue 11-14, Biblioteca storica,

Fri 14-17 (14-16: Biblioteca Storica, 16-17: 29A).

Introduction:

This is an introductory course to Algebraic Geometry, aimed at students whose only background are the courses of Tor Vergata's Laurea Triennale in Mathematics. The plan is to follow Joe Harris' book "Algebraic Geometry. A first course" (henceforth referred to as AG) and go through as many chapters as we can. Especially, lots of exercises will be solved (mostly by the students).

Week 1. mar 1-4 (only thursday and friday)

Affine and projective space. Affine and projective varieties. Linear spaces. Finite sets. Projectivities (PGL(n,k)).

Reference : AG up to page 8 (except Th. 1.4).

Assignements : Ex. 1.3, 1.6. Prove the assertions at the last line lines of p.7 and first lines of p.8 on the cross-ratio.

Week 2. mar 7 - 11

Twisted cubic and rational normal curves. Zariski topology. Regular functions on affine varieties. Correspondence affine varieties vs finitely generated k-algebras without nilpotents. Maximal ideals vs points. Regular functions on projective varieties. Morphism to affine varieties. Morphisms to projective spaces. Veronese maps and Veronese varieties. Products of paffine varieties. products of projective spaces.

Reference : AG Chapter 1 until 1.21 (included). AG Chapter 2 until 2.11 (included)

Assignements : Exercises in AG concerning the above topics.

Week 3. mar 14 - 18 (only friday from 2 -to 6)

Products of projective spaces, and their subvarieties, bihomogeneous polynomials. Categorical product. Graph of a morphism (is closed). Categorical fibered product.

Cones. Quadrics. Projections. Resultant of two polynomials. Theorem: the image of a projective subvariety via a projection is a projective subvariety.

Reference : AG Chapter 2 (all). AG Chapter 3 until 3.9 (included)

Assignements : Exercises in AG concerning the above topics.

Week 4, mar 21- 25 (only tuesday)

Image of a projective subvariety via a morphism, and consequences. Examples. Image of a conctructible set via a morphism. Reference : AG Chapter 3 from 3.10 to the end. AG chapter : 4.9 and 4,10

Assignements : Exercises in AG concerning the above topics. Week 5. mar 28 - apr 1. (only friday)

Nullstellensatz and consequences. Varieties defined by a set of equations set-theoretically, ideal-theoretically, locally. Saturation of an ideal. Reference : AG Chapter 5 (all)

Assignements : Exercises in AG concerning the above topics. Week 6. apr 4 - apr 7. (only friday)

Grassmannians. Plucker equations.

Reference : AG Chapter 5 (all)

Assignements : Exercises in AG concerning the above topics.