ALGEBRAIC GEOMETRY

The course aims to provide an introduction to the basic theories and techniques in modern Algebraic Geometry.

Besides the theoretical lectures developing the foundations of the subject there will be numerous exercise sessions,

where the students will solve at the blackboard exercises from preassigned lists and work out the properties of

relevant explicit examples in order to have a solid basis of objects on which to test the abstract theories.

I) -- Affine and projective algebraic sets. Zariski topology on affine and projective spaces. Correspondence between affine algebraic sets

and radical ideal in a polynomial ring (algebraically closed field). Irreducible algebraic sets and correspondence with prime ideals.
Coordinate ring of an affine algebraic variety and of a projective variety. Decomposition of an algebraic set into irreducible components
and its relations with primary decomposition of an ideal. Dimension of an algebraic variety: topological and algebraic definition.

II) -- Regular functions on a quasi-projective variety: definition and first properties. Examples and applications. Morphisms between varieties: definition and first properties.
Examples and applications. Local ring of regular functions on a variety: definition and first properties. Rational functions on a variety: definition and first properties.
Rational (and birational) maps between algebraic varieties: definitions and first examples. Correspondence between dominant rational maps and homomorphisms of their
function fields. Regular functions on a projective varieties and applications.

III) -- Product of algebraic varieties: universal property, existence and unicity. Examples and applications: graph morphism, diagonal morphism, decomposition of a morphism via
its graph morphism and projections. Fundamental Theorem of Elimination Theory. Examples and applications.

IV) -- Non-singular point on an algebraic variety: extrinsic and intrinsic definition. Singular locus. Blow-up of a variety at a point. Tangent cone and tangent space to a variety at a point: extrinsic and intrinsic definition. Examples and applications. Definition of multiplicity of a point on a variety. Comparison between the tangent cone and the tangent space at a point: non-singularity criterion.

V) -- Theorem on the dimension of the fibers of a morphism. Applications. Irreducibility Criterion. Applications to the study of lines on superfaces in projective space with special regard to the case of cubics. Dual variety and Bertini Theorem.

VI) -- Intersection in projective spaces. Dimension of the intersection of two projective varieties. Degree of a projective variety: geometrical definition and algebraic definition via Hilbert Polynomial. Revision on the annihilator and the associated primes of a graded module. Multiplicity of a module along a minimal prime. Hilbert-Serre Theorem and Hilbert Polynomialof a projective variety. Generalized B\' ezout Theorem and applications. Local definition of intersection multiplicity and comparison with the multiplicity of an associated module. Intersection multiplicity of two plane curves; examples and first properties. Flex of plane algebraic curves and Hessian curve. Examples and applications. Study of some classes of singular points: ordinary multiple points e their resolution via blow-up's. Non-ordinary singularities and tacnodes.

00) M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin, Letture su curve, superficie e varietà proiettive speciali. Un' introduzione alla Geometria Algebrica, Bollati Boringhieri.

0) R. Hartshorne, Algebraic Geometry, Springer Verlag.

1) W. Fulton, Algebraic Curves--An Introduction to Algebraic Geometry, http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

2) I. Dolgachev, Classical Algebraic Geometry, http://www.math.lsa.umich.edu/~idolga/CAG.pdf

3) I. R. Shafarevich, Basic Algebraic Geometry, Springer-Verlag.

4) D. Mumford, The Red Book of Varieties and Schemes, Springer Verlag.