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Shafarevich Springer.

Too Many Topics

Algebraic geometry I: complex projective varieties by D. Mumford Springer. Curves and their Jacobians by D. The red book of varieties and schemes by D.

Complex algebraic curves by F. Kirwan Cambridge University. Quelques aspects de la surfaces de Riemann by E. Reysatt Birkhauser. Waldschmidt articles by J. Bost, etc Springer. Algebraic varieties by G. Kempf London Math Soc. Algebraic geometry: introduction to schemes by I. Macdonald Benjamin. Introduction to commutative algebra and algebraic geometry by E. Kunz Birkhauser. Geometry of schemes by D.

Niranjan Ramachandran

Eisenbud and J. Harris GTM. Algebraic geometry: a first course by J. Undergraduate algebraic geometry by M. Gottsche et al AMS. Methods of algebraic geometry by W.

Algebraic Geometry - Lothar Göttsche - Lecture 01

Hodge and D. Pedoe Cambridge Math Library.


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April: The link of a singularity, Newton polygon, Puiseux' theorem. How to see singular points on algebraic curves, even without a magnifying glass, by deforming the equation. Beginning to study the local topology of an algebraic curve in the neighbourhood of a singular point. Definition of weighted-homogeneous polynomials, Newton polygon of a polynomial f x, y.

Definition of Newton series, Puiseux series, Newton pairs, Puiseux pairs.

May: Newton's algorithm. May: Knots, links, Seifert surfaces. Brief introduction to knot theory: knots and links in S 3 , isotopy, diagrams, positive and negative crossings, Seifert surfaces. Every link has a Seifert surface: Frankl-Pontrjagin's proof and Seifert algorithm. Linking numbers: various definitions. May: Cable knots.

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Algebraic Curves in Cryptography - CRC Press Book

Torus knots, meridian and preferred longitude of an embedded torus in S 3 , definition of the a, b -cable of a knot K, denoted K a, b. May: Conical structure theorem, positive braids, Milnor's fibration. Definition of the braid group on n strands as the fundamental group of the space of polynomials with n distinct roots; definition of positive braids. Theorem: Links of singularities are positive braids.

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Proof sketch using the description of the link of a singularity as an iterated cable of the unknot. Definition of locally trivial fibre bundles, definition of fibred links. Statement of Milnor's fibration theorem links of singularities are fibred: the argument map of the defining polynomial is a locally trivial fibration. First examples and illustrations: the Hopf link, elastic bands.

Algebraic Curves, Algebraic Manifolds and Schemes

June: Repetition session. Seifert surface of genus zero for the link in exercise 22, Seifert's algorithm applied to the same link gives a surface of genus two. The p i can become equal to one and go back to numbers strictly bigger than one, as it happens for instance in exercise 21 b. Links of singularities are iterated cables of the unknot: details on why the link of a singularity is an iterated cable comparison of the radii of the tori on which the cables lie.

Meridian and longitude; drawing of the 2, 13 -cable of the trefoil knot T 2, 3. June: Fibration and monodromy of algebraic knots. Theorem: Algebraic knots are fibred. Proof by showing that cables of fibred knots are fibred. Monodromy of a fibred knot as the glueing map of a mapping torus and as the flow of a vector field.

July: Seifert form, Alexander polynomial, homological monodromy. Homology of a surface. Definition of the Seifert form associated to a Seifert surface, definition of the Alexander polynomial of a knot. Theorem: The Alexander polynomial of a fibred knot equals the characteristic polynomial of its homological monodromy proof using the monodromy flow.

Consequence: the Alexander polynomial of a fibred knot is monic and of degree 2g, where g is the genus of its fibre surface. July: Blowing up, resolution of singularities and the monodromy. Idea of blowing up: polar coordinates.