Lecture

Plane Curves: Singular Points and Multiplicities

Related lectures (32)

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Covers affine algebraic sets, hypersurfaces, elliptic curves, ideals, and Noetherian rings in algebraic geometry.

Explores affine varieties, hypersurfaces, dimension in algebraic geometry, minimal prime ideals, and local properties of plane curves.

Local Rings and Residues

Covers the proof of theorem 4.2 on multiplicities and the special structure of local rings at a simple point of a plane.

Dedekind Rings: Integral Extensions and Noetherian Rings

Explores Dedekind rings, integral extensions, and noetherian rings in algebraic structures.

Ring Operations: Ideals and Classes

Covers the operations in rings, ideals, classes, and quotient rings.

Irreducible Factors and Noetherian Rings

Explores irreducible factors, Noetherian rings, ideal stability, and unique factorization in rings.

Dimension Theory of Rings

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Projective Plane Curves

Introduces projective plane curves, degrees, components, multiplicities, intersection numbers, tangents, and multiple points, culminating in the statement of Bézout's theorem and its consequences.

Prime ideals and local rings

Covers the concept of prime ideals and local rings, emphasizing their importance.

Principal Ideal Domains: Structure and Homomorphisms

Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.

Dedekind Rings and Fractional Ideals

Explores Dedekind rings, fractional ideals, integrally closed properties, prime ideal factorization, and the structure of fractional ideals as a commutative group.

Curvature and Inflection Points

Explores curvature, inflection points, and angular functions in plane curves, highlighting the importance of inflection points.

Commutative Algebra: Recollections

Covers fundamental concepts in commutative algebra, including rings, units, zero divisors, and local rings.

Curves with Poritsky Property and Liouville Nets

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

Ideals and Representations

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Polynomials: Theory and Operations

Covers the theory and operations related to polynomials, including ideals, minimal polynomials, irreducibility, and factorization.

Ideals and PPCM

Covers the concept of ideals in polynomial rings and their properties.

Irreducible Algebraic Sets

Explores irreducible algebraic sets and their unique decomposition into components, with a focus on prime ideals and plane subsets classification.

Projective Geometry: Tangent Lines

Discusses the correction of tangent lines and projective plane curves, exploring topological applications and curve structure.