Algebraic Geometry 2

ALGEBRAIC GEOMETRY 2

Undergraduate and Graduate studies in Mathematics

Roma Tre University

A.Y. 2018/2019

Docente: Angelo Felice Lopez

Teacher: Angelo Felice Lopez

SCHEDULE OF LESSONS: Tuesday 11-13 and Wednesday 14-16 in room 009.

FIRST LESSON: Tuesday February 26, 2019.

GENERAL DESCRIPTION

Algebraic Geometry is the study of algebraic varieties, that is sets of common zeros of some polynomials. Historically this study is performed by the analysis of geometrical, algebraic, topological, differential, analytic and numerical properties. Such richness of points of view makes Algebraic Geometry one of the most fascinating and central areas of mathematics. Many famous problems in mathematics, like Fermat's Last Theorem, have been solved with the essential use of Algebraic Geometry.
Classically basic properties of affine and projective algebraic varieties were studied. But, starting from the 60's, it was clear that this was no longer sufficient for deeper and more rigorous studies and for applications to number theory, physics or other fields. For this reason some famous mathematicians like Weil, Serre and especially Grothendieck, introduced the theory of schemes and cohomology. These, that nowadays are to be considered basic notions in Algebraic Geometry, are the subjects that will be studied in the course.

OUTLINE OF CONTENT

Sheaf theory and its use in schemes	Presheaves and sheaves, sheaf associated to a presheaf, relation between injectivity and bijectivity on the stalks and analogous properties on the sections. The category of ringed spaces. Schemes. Examples. Fiber products. Algebraic sheaves on a scheme. Quasi-coherent and coherent sheaves.
Sheaf cohomology	Homological algebra in the category of modules over a ring. Flasque sheaves. Cohomology of sheaves using the canonical resolution with flasque sheaves.
Cohomology of quasi-coherent and coherent sheaves on a scheme	Cech cohomology and ordinary cohomology. Cohomology of quasi-coherent sheaves on an affine scheme. The cohomology of the sheaves O(n) on the projective space. Coherent sheaves on projective space. Euler-Poincaré characteristic.
Invertible sheaves and linear systems	Glueing of sheaves. Invertible sheaves and their description. The Picard group. Morphisms in a projective space. Linear systems. Base points. Ample and very ample linear systems and sheaves. Criteria for ampleness.

Final contents $[pdf]$

SUGGESTED BOOKS:

We will mainly use the notes of the course written by E. Sernesi (here). Update: these notes have been reworked here.

We also suggest the following classical books:

* R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
* D. Eisenbud, J. Harris: The Geometry of Schemes, Springer Verlag (2000).
* U. Gortz, T. Wedhorn: Algebraic Geometry I, Viehweg + Teubner (2010).

and the following

Algebra books:

* M. Artin, Algebra, Pearson 1991.
* M.F. Atiyah, I.G. Mac Donald, Introduction to commutative algebra, Addison-Wesley 1969.