BARTH'S SEXTIC UBIQUITY OF ALGEBRAIC GEOMETRY BLOW UP OF A PLANE
                                                                                                                               

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ALGEBRAIC GEOMETRY 1
Undergraduate and graduate studies in Mathematics
Roma Tre University
A.Y. 2022/2023 
                  

Teachers: Angelo Felice Lopez and Filippo Viviani.

SCHEDULE OF LESSONS: Monday 12PM-2PM, Tuesday 2PM-4PM, room M4. Friday 10AM-12PM, room M4 (to be organized).

FIRST LESSON: to be announced, room M4.


DESCRIPTION OF THE COURSE

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.
The course introduces to basic properties of affine and projective algebraic varieties, to maps among them, to their local geometry and to the theory of divisors and linear systems.

OUTLINE OF CONTENT
Affine spaces
 Zariski topology. Affine closed subsets and radical ideals. Irreducible components. Regular functions and morphisms. Examples. Finite morphisms.
Varieties Projective spaces. Quasi-projective varieties. Projective hypersurfaces. Rational and regular functions. Morphisms. Affine varieties. Dimension. Generically finite morphisms. Constructible sets. Birational equivalence.
Geometry in projctive spaces Rational normal curves. Veronese varieties. Products. Projections. Invariance of projective closure. Complete intersections. Upper semicontinuity of dimension.

If there will be time, we will also do: local geometry, normalization; divisors, linear systems and associated morfisms.


Final program  [pdf]

SUGGESTED BOOKS:

We will follow closely the lecture notes written by L. Caporaso (the notes will be given in class).

We also suggest the following classical textbooks:
* R. Hartshorne, Algebraic geometry, Graduate Texts in Math. No. 52. Springer-Verlag, New York-Heidelberg, 1977.
*  I. Shafarevich, Basic algebraic geometry vol. 1, Springer-Verlag, New York-Heidelberg, 1977.
* J. Harris, Algebraic geometry (a first course), Graduate Texts in Math. No. 133. Springer-Verlag, New York-Heidelberg, 1977.

and

Commutative Algebra textbooks:

* M. Artin, Algebra, Pearson Education 2014.
* M.F. Atiyah, I.G. Mac Donald, Introduction to commutative algebra, Sarat Book House, 1996.