Module MATH4011: ALGEBRAIC GEOMETRY IV

Department: Mathematical Sciences

MATH4011: ALGEBRAIC GEOMETRY IV

Prerequisites

• Complex Analysis II (MATH 2011) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Actuarial Mathematics II (MATH2171) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Symmetries II (MATH2111) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Hyperbolic Geometry II (MATH2121) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Probability II (MATH2**1) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061).

Corequisites

• None.

Excluded Combination of Modules

• Algebraic Geometry III (MATH3321).

Aims

• To introduce the basic theory of algebraic varieties and birational geometry, with particular emphasis on plane curves.

Content

• Affine and projective plane curves over a field k.
• Conics, Pappus' Theorem.
• Unique factorisation in polynomial rings.
• Study's lemma, irreducibility.
• Singular points, tangents, points of inflection.
• Dual plane, linear systems of curves.
• Bezout's theorem: Resultants, weak form of Bezout, applications of Pascal's theorem, Cayley-Bacharach theorem, group law on a cubic. Intersection multiplicity, strong form of Bezout.
• Bezout's theorem: applications, flexes, Hessian, configuration of flexes of a cubic.
• Elliptic curves, Weierstrass normal form.
• Complex curves as real surfaces.
• Basic topology and manifolds.
• Degree-genus formula.
• Resolution of singularities and non-singular models.
• Reading material on a topic in the following area:parametrizing the branches of a curve by Puiseux Series.

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Algebraic Geometry.
• have an understanding of specialised and complex theoretical mathematics in the field of Algebraic Geometry.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Affine and projective plane curves.
• Unique factorisation in polynomial rings.
• Linear systems.
• Bezout's theorem and applications.
• Topology of curves over the complex numbers.
• Knowledge and understanding of a topic in parametrisation of branches of a curve.

• Students will have highly specialised and advanced mathematical skills which will be used with minimal guidance in the following areas: spatial awareness, abstract reasoning.
• Students will have the ability to read independently to acquire knowledge and understanding in the area of branch parametrisation.

• Students will have enhanced problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University