Algebraic Geometry I

G. Occhetta, L.E. Solá Conde and J. Wisniewski

Flag bundles on Fano manifolds,

to appear in Journal de Mathématiques Pures et Appliquées

G. Occhetta, L.E. Solá Conde and K. Watanabe

Uniform families of minimal rational curves on Fano manifolds,

to appear in Revista Matematica Complutense

R. Muñoz, G. Occhetta, L.E. Solá Conde, K. Watanabe and J. Wisniewski

A survey on the Campana-Peternell Conjecture

Rendiconti dell'Istituto di Matematica dell'Università di Trieste 47, (2015) 127-185

G. Occhetta, L.E. Solá Conde and K. Watanabe and J. Wisniewski

Fano manifolds whose elementary contractions are smooth P1-fibrations

to appear in Annali Sc. Norm. Super. Pisa Cl. Sci.

R. Muñoz, G. Occhetta, L.E. Solá Conde and K. Watanabe

Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle.

Math. Ann., 361, (2015), 583-609

R. Muñoz, G. Occhetta and L.E. Solá Conde

A classification theorem for Fano bundles.

Annales Inst. Fourier 64 (2014), no. 1, 341-373.

R. Muñoz, G. Occhetta and L.E. Solá Conde

On rank 2 vector bundles on Fano manifolds.

Kyoto Journal of Mathematics 54 (2014), no. 1, 167-197.

C. Novelli, G. Occhetta

Manifolds covered by lines and extremal rays.

Canad. Math. Bull. 55 (2012), no. 4, 799–814.

R. Muñoz, G. Occhetta and L.E. Solá Conde

Rank two Fano bundles on G(1,4).

Journal of Pure and applied Algebra, 216, (2012), 2269-2273

G.Occhetta, V. Paterno

Rationally cubic connected manifolds II.

Revista Mathemática Iberoamericana, 28, (2012), 815-838

G.Occhetta, V. Paterno

Rationally cubic connected manifolds I: manifolds covered by lines.

J. Math. S

Riemann Surfaces

Definition of a Riemann surface and first examples: Riemann sphere, projective line, complex tori, smooth plane curves, affine and projective. Space curves.

Holomorphic and meromorphic functions

Definitions and examples. A global holomorphic function on a compact Riemann surface is constant. Characterization of meromorphic functions on the Riemann sphere and the projective line. Meromorphic functions on complex tori and plane curves

Holomorphic maps

Definition. Examples. Identity principle and open mapping theorem and corollaries. Local normal form and multiplicity. Degree of holomorphic maps. Applications. Meromorphic functions on a complex torus.

More examples of Riemann surfaces

Lines and conics. Hyperelliptic Riemann surfaces. Morphisms of complex tori.

1-forms

Holomorphic and meromorphic 1-forms. Examples. Pullbacks

Divisors

Definitions. Principal and canonical divisors. Pullbacks. Riemann-Hurwitz formula and applications. Plane curves: Bezout's Theorem and Clebsch formula.

Linear equivalence: definition, examples and basic properties. Principal divisors for the Riemann sphere and the complex tori.

Riemann-Roch spaces

Definition, first properties, dimension for Riemann-Roch spaces on the Riemann sphere. Linear systems

Riemann-Roch spaces on complex tori. Riemann-Roch spaces are finite dimensional. Base points of linear systems.

Maps to projective space

Base point free linear systems. Very ample divisors. The hyperplane divisor.

Examples: Rational normal curves. The elliptic cubic.

Projective geometry

Review of basic projective geometry. Projection of smooth curves from an outside point.

Riemann Roch Theorem

Statement and first consequences.

Canonical curves

Base point freeness of the canonical linear system for positive genus. Hyperelliptic curves. Canonical map for hyperelliptic curves. Canonical curves of genus 3 and 4. Geometric form of Riemann-Roch. Canonical curves of genus five.

Miscellaneous

Clifford's Theorem

Castelnuovo's bound on the genus of a space curve. Gap numbers of a linear system. Wronskian criterion. Flexes of a plane curve.

Affine varieties

Basic notions from commutative algebra. The Zariski topology on A^n. Affine algebraic sets and affine varieties. Irreducibility. Dimension. Coordinate rings.

Projective varieties

Definitions. Projective closure. Morphisms. Coordinate rings.

Quasi projective varieties

Definitions. Morphisms. Affine basis. Regular functions. Examples: Veronese and Segre embe

Published papers

ALGEBRAIC GEOMETRY I

List of topics