2019/2020

2019/2020

DM270

MAT/03 (GEOMETRY)

DEPARTMENT OF MATHEMATICS "FELICE CASORATI"

MATHEMATICS

PERCORSO COMUNE

1°

2nd semester (02/03/2020 - 09/06/2020)

6

48 lesson hours

Italian

ORAL TEST

FREDIANI PAOLA (titolare) - 3 ECTS
NARANJO DEL VAL JUAN CARLOS - 3 ECTS

Linear algebra and some knowledge on affine and projective geometry and basic abstract algebra (rings and fields).

We would like to give the basic notions on affine and projective varieties. Many examples, exercises and applications of the theory will be discussed.

Affine and projective varieties: Zariski topology, irreducible sets and prime ideals, noetherianity. Affine and projective Nullstellensatz. Regular and rational maps on a variety. Field of rational functions. Morphisms. Dominant and rational maps, birational equivalence. Cremona and Veronese maps. Elimination theory: Products, Segre embedding. Elimination theorem.
Dimension: Dimension of a variety as the trascendence degree of the function field. Dimension of a product. Krull dimension and Hauptidealsatz. Dimension of the fibers of maps. Blow- ups. Grassmannians. Plu ̈cker coordinates. Equations of Grassmannians. Lines on surfaces in P3.
Local theory: Intersection of lines and hypersurfaces. Embedded tangent space and Zariski tangent space. Functoriality. Jacobian criterion. Singularities. Some topics on plane curves (Intersection multiplicity, Bezout, Plane Cubics,...).

Lectures and exercise sessions.

1. Harris, J., Algebraic Geometry : a first course. New York ; Springer, 1992.
2. Hartshorne, R., Algebraic geometry. New York : Springer, 2000.
3. Hasset, B. Introduction to Algebraic Geometry. Cambridge : Cambridge University, 2008.
4. Hulek, K., Elementary Algebraic Geometry. Providence [R.I.] : American Mathematical Society, 2003.
5. Looijenga, E., A first course on Algebraic Geometry (electronic text: http://www.staff.science .uu.nl/ looij101/AG2016.pdf).
6. Shafarevich, I. R., Basic algebraic geometry, Berlin : Springer, 1994, 2nd ed.
7. Mumford, D., The Red book of varieties and schemes, Berlin: Springer, 1999.
8. Smith, K., Kahanp ̈a ̈a, L., Kek ̈al ̈ainen, P., Traves,W., An Invitation to Algebraic Geometry, Uni- versitext, Springer Verlag, New York, 2000.

Oral exam. We will verify both the knowledge of the theory and the ability to solve problems and exercises.