ALGEBRAIC GEOMETRY

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

I) -- Revision on projective spaces: parametric and cartesian equations of linear spaces, dual projective space. Linear systems of hypersurfaces and applications to the first enumerative problems with special regards to conics and hyperquadrics. Lines incident four general lines in the space. Klein quadric as parameter space of lines in projective space. Planes in the Klein's quadric via the geometry of lines.

II) -- 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.

III) -- 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.

IV) -- 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.

V) -- 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.

VI) -- 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. Secant and plurisecant spaces to a variety: definitions and examples. Gauss map: definition and examples.

VII) -- 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 Polynomial

of 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.

It time allows some topics will be chosen between the following ones:

0) -- Tensor product of vector spaces: universal property, existence and unicity. Some canonical isomorphisms. Constructions of Segre varieties and study of their geometrical properties. Multilinear applications and tensor product. Tensor algebra of a vector space. Symmetric algebra of a vector space. Construction of Veronese varieties and study of their geometrical properties. Antisymmetric tensor algebra of a vector space. Determinants and generalized Bined and Laplace formulas. Decomposable vectors and Grassmann varieties. Tensor product of algebras over a field. Examples and applications.

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.